# Nonlinear Phenomena of Ultracold Atomic Gases in Optical Lattices: Emergence of Novel Features in Extended States

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## Abstract

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## 1. Introduction

## 2. Theoretical Framework

#### 2.1. Setup of the System

#### 2.2. Bosons

#### 2.3. Fermions

#### 2.4. Discrete and Continuum Models

#### 2.5. Energetic and Dynamical Stability

## 3. Swallowtail Loops in Band Structure

#### 3.1. Basic Physical Idea: The Nonlinear Landau–Zener Model and Variational Ansatz for Condensate Wavefunction in Optical Lattices

#### 3.2. Swallowtail Loops Structures for Bosons in Optical Lattices

#### 3.2.1. Occurrence of Loop Solutions

#### 3.2.2. Stability of Loop Solutions

#### 3.3. Experimental Realization

#### 3.4. Other Extensions

#### 3.5. Future Prospects

## 4. Multiple Period States in Cold Atomic Gases in Optical Lattices

#### 4.1. Basic Physical Idea: A Simple Explanation of the Emergence of Multiple Period States by a Discrete Model

#### 4.2. Multiple Period States in BECs

#### 4.3. Multiple Period States in Superfluid Fermi Gases

## 5. Nonlinear Lattices

#### 5.1. Dynamical Stability of the Superfluid: Special Properties of Nonlinear Lattices

#### 5.2. Basic Physical Idea: The Dynamical Stability of Nonlinear Lattices

#### 5.3. Superfluid Cold Atomic Gases in Nonlinear Lattices

#### 5.4. Experimental Setup

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**) Adiabatic energy levels (or the chemical potential μ) as a function of the nonlinearity c for the nonlinear LZ problem showing the emergence of loops for $c>v$. The dashed lines represent the adiabatic energy levels for the $c=0$ case. (

**Right**) Tunneling probability as a function of drive speed α at different values of c. For the linear case the result of the classic formula ${r}_{0}=exp(-\pi {v}^{2}/2\alpha )$ is displayed by the open circles. The figures are taken from [34].

**Figure 2.**(

**Top**) Swallowtail loop structure of the energy per particle as function of k computed from the variational ansatz (24) for ${V}_{0}=3\hslash {(4{V}_{0}/m)}^{1/2}\pi /d$ and $n=1.2{V}_{0}/{U}_{0}$. Figure was taken from [19]. (

**Bottom**) Energy per particle obtained using the variational ansatz (23) demonstrating the presence of swallowtail loops at both the zone edge and zone center for sufficiently strong nonlinearity. ${U}_{0}$ in the figure is g in our notation. Moreover, the loops persist and extend over the entire band in the limit ${V}_{0}\to 0$. Taken from [32].

**Figure 3.**(

**a**) The dashed line schematically shows the energy dispersion for a free particle which is modified to the solid line showing the (lowest) band structure on the application of a periodic potential of period L. (

**b**) Schematic plot of energy bands for a superfluid in an optical lattice that screens out the lattice to maintain non-zero flow velocity at the zone edge. Plots taken from [35].

**Figure 4.**Contour plot of width of swallowtail loops from a numerical minimization using the variational wavefunction (11) as a function of interaction ($n{U}_{0}$ is $ng$ in the notation of this review) and optical lattice depth ${V}_{0}$ for zone-edge loops (

**a**) and zone-center loops (

**b**). Dashed lines in (a) are for truncation with ${l}_{\mathrm{max}}=2$ and solid lines for ${l}_{\mathrm{max}}=3$. Taken from [32].

**Figure 6.**(

**a**) Schematic illustration of the physical principle behind the definitions of energetic and dynamical stability. A dynamically unstable state is always energetically unstable whereas the vice-versa is not true. (

**b**) Stability phase diagram at different values of lattice depth and nonlinearity parameter for a BEC in an optical lattice. The relation between the notation in the picture and this review article is scaled potential $v={V}_{0}/\left(8{E}_{0}\right)$ and nonlinearity parameter $c=gn/\left(8{E}_{0}\right)$. q is the wave number of the perturbation modes and k the quasimomentum in units of $2K=4\pi /d$. In the shaded dark and light areas the system is energetically unstable while in the shaded dark area it is dynamically unstable. Triangles in (a.1–a.4) represent the boundary ${q}^{4}+c={k}^{2}$ expected for $v=0$ and full dots in first column are from the results of an analytical calculation using Equation (31) valid for $v\ll 1$. Broken curves indicate the most unstable modes for each quasimomentum. The open circles in (b.1) and (c.1) represent analytical result valid for $c\ll v$ (see Equation (5.14) of [43]). Figure taken from [43].

**Figure 7.**Contour plot of maximum quasimomentum k for energetic stability (

**a**) and dynamical stability (

**b**) as a function of interaction and lattice depth for zone-edge loops. $k\le \pi /d$ represents states on the lowest branch of the ground band and $k>\pi /d$ the states on the lower edge of the swallowtail loop. The dashed lines in (

**b**) are energetic stability curves for comparison. Taken from [32].

**Figure 8.**(

**Left**) (

**a**) Transfer efficiency ${n}_{R}$ from filled to unfilled well (tube) as a function of the energy detuning for ground state sweep (gray dots) and excited state sweep (red dots) for constant sweep rate of $2\pi {J}^{2}/\hslash \left|\alpha \right|=2.1\left(1\right)$; (

**b**) Adiabatic energy levels for the excited (with loops) and ground state levels including strong repulsive interaction (solid lines) and excluding interactions (dashed lines). The curved lines indicate the relaxation process enabling non-adiabatic transitions. $|{n}_{R},{n}_{L}\rangle $ is a shorthand denoting the left- and right-well (tube) population respectively. (

**Right**) (

**a**) Transfer efficiency as a function of z-lattice depth at constant sweep rate $0.53\left(3\right)$; (

**b**) Phase diagram of the metastable excited condensate branch. In the gray shaded area there is a loop in the adiabatic energy level. Data points represent minima in the measured transfer efficiency and agree well with the solid line depicted for the calculated maximum loop size. Taken from [64].

**Figure 9.**(

**Left**) (

**a**) Schematic plot of the energy landscape of a hysteretic system as a function of the applied field F (for example Ω the rotation rate of the superfluid) showing stable states of different energies separated by a barrier; (

**b**) Energy diagram for a superfluid as a function of Ω showing a swallowtail loop and the related hysteresis loop; (

**c**) The swallowtail structure is periodic in the rotation quanta ${\Omega}_{0}$. (

**Right**) (

**a**–

**f**) Measured hysteresis loop with sigmoid fits at different values of potential strength ${U}_{2}$ shown in (

**g**). The red up-triangles (blue down-triangles) are results from 20 shot averages while starting from $n=0$ ($n=1$). (

**g**) Experimentally determined size of the hysteresis loop (green dots) as a function of ${U}_{2}$ and open and filled cyan diamonds are results of numerical GP calculation of dynamics with differing amounts of phenomenological dissipation. Taken from [39].

**Figure 10.**Energy per particle ϵ as a function of k for period-1 (blue dashed lines) and period-2 (red solid lines) states of BECs in a periodic potential obtained from the discrete model, for (

**a**) $U\nu /2K=1/2$; (

**b**) $U\nu /2K=1$; (

**c**) $U\nu /2K=2$. In the case of the (

**a**), period-2 states exist in the limited region of $1/6\le kd/2\pi \le 1/3$.

**Figure 11.**Energy per particle $E/n$ of BECs in a periodic potential as a function of k for the lowest bands obtained from the GP equation for the continuum model. The bands of the period-2 states are shown by the thick solid lines. In the notations of the present article, ${U}_{0}=g$ and ${V}_{0}=s{E}_{R}/2$. This figure is taken from [33].

**Figure 12.**Density profiles ${\left|\psi \left(x\right)\right|}^{2}$ of period-2 states (upper panel) of BECs and the periodic external potential $V\left(x\right)$ (lower panel). The solid (dashed) line in the upper panel shows ${\left|\psi \left(x\right)\right|}^{2}$ for the lower-energy (higher-energy) period-2 state of Figure 11a at $kd/2\pi =1/4$, $gn={E}_{R}$, and $s=0.2$. This figure is taken from [33].

**Figure 13.**Profiles of (

**a**) the magnitude of the pairing field $|\Delta (x\left)\right|$ and (

**b**) the density $n\left(x\right)$ of the lowest period-2 states of superfluid Fermi gases in the BCS-BEC crossover: $1/{k}_{F}{a}_{s}=-1$ (red solid line), $-0.5$ (green dashed line), and 0 (blue dotted line). The quasimomentum P of the superflow is set at the Brillouin zone edge $P={P}_{\mathrm{edge}}/2=\hslash {q}_{B}/4$ of the period-2 states, and other parameters are set at $s=1$ and ${E}_{F}/{E}_{R}=0.25$. This figure is adapted from [82].

**Figure 14.**Energy E per particle of superfluid Fermi gases in a periodic potential as a function of the quasimomentum P. Parameter values are $s=1$, ${E}_{F}/{E}_{R}=0.25$, and $1/{k}_{F}{a}_{s}=-1$. The normal Bloch states with period d are shown by the blue dotted line with • symbols, and the period-2 states are shown by the red solid line with +. Note that the period-2 states are energetically more stable than the normal Bloch states in the region of $0.2\lesssim P/{P}_{\mathrm{edge}}\le 0.5$.

**Figure 15.**Difference $\Delta E\equiv {E}_{2}-{E}_{1}$ of the total energy per particle between the period-2 states (${E}_{2}$) and the normal Bloch states (${E}_{1}$) at $P={P}_{\mathrm{edge}}/2$ along the BCS-BEC crossover. The red solid line with + is for $s=1$ and the blue solid line with × is for $s=2$; ${E}_{F}/{E}_{R}=0.25$. The green dashed line shows the results by the GP equation for parameters corresponding to $s=1$ and ${E}_{F}/{E}_{R}=0.25$. This figure is taken from [82].

**Figure 16.**Growth rate γ of the fastest growing mode (black solid line) and survival time ${\tau}_{\mathrm{surv}}$ of the period-2 state at $P={P}_{\mathrm{edge}}/2$ ($s=1$ and ${E}_{F}/{E}_{R}=0.25$). Blue dashed-dotted, green dotted, magenta dashed double-dotted, and red dashed lines show ${\tau}_{\mathrm{surv}}$ for relative amplitude $\tilde{\eta}\left(0\right)$ of the initial perturbation of $10\%$, $1\%$, $0.1\%$, and $0.01\%$, respectively. This figure is taken from [82].

**Figure 17.**Density distributions in the lowest band of the normal Bloch states (i.e., period-1 states whose period is one supercell) as functions of k for different values of $U\nu /2K$. Panels (

**a**,

**b**): Populations of $|{g}_{1}{|}^{2}$ (attractive site) and $|{g}_{2}{|}^{2}$ (repulsive site) for $U\nu /2K=6$, respectively. Panels (

**c**,

**d**): Populations of $|{g}_{1}{|}^{2}$ and $|{g}_{2}{|}^{2}$ for $U\nu /2K=0.75$, respectively. This figure is taken from [86].

**Figure 18.**Dynamical stability diagrams for the normal Bloch states (i.e., period-1 states) for $U\nu /2K=6$ (

**a**) and $0.75$ (

**b**). Quasi-wave numbers k and q are in units of $\pi /2\tilde{d}$. The white regions are the dynamically unstable regions and the gray-shaded regions are the dynamically stable regions. The contours show the growth rate of the fastest growing mode, i.e., the largest maximum absolute value of the imaginary part of the eigenvalues of matrix ${\sigma}_{z}M\left(q\right)$ in Equation (17) in units of K. This figure is taken from [86].

**Figure 19.**Stability diagrams in nonlinear lattices for various values of ${c}_{1}\equiv mn{V}_{1}/\left(4{\hslash}^{2}{k}_{0}^{2}\right)$ and ${c}_{2}\equiv mn{V}_{2}/\left(4{\hslash}^{2}{k}_{0}^{2}\right)$, where n is the average number density. The quasi-wave numbers k and q are in units of $2{k}_{0}=2\pi /d$ with d being the lattice constant of the unit cell in the notation of this review. The Bloch states are stable in the white area. In the gray area the Bloch states are energetically unstable but dynamically stable while, in the black area, they are unstable both energetically and dynamically. This figure is taken from [85].

**Figure 21.**Phase diagram of the Bloch states of superfluid Fermi gases in nonlinear lattices in the $|{U}_{1}|$–${U}_{2}$ plane, where ${U}_{1}={V}_{1}$ and ${U}_{2}={V}_{2}$ in the notation of this review. In the parameter region denoted by “SF-N”, the system has a critical quasimomentum of the superflow above which the normal state has lower energy compared to the superfluid state while, in the regions denoted by “SF” and “SW”, there is no such critical value and the superfluid state is always lower in energy than the normal state in the whole Brillouin zone. Particularly, in the region of “SW” at larger $|{U}_{1}|/{U}_{2}$, the energy band has a swallowtail loop around the zone edge. In the region denoted by “FFLO-like” (Fulde–Ferrell–Larkin–Ovchinnikov) at smaller $|{U}_{1}|/{U}_{2}$, the ground state has a nonzero quasimomentum of the superflow. Taken from [89].

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Watanabe, G.; Venkatesh, B.P.; Dasgupta, R.
Nonlinear Phenomena of Ultracold Atomic Gases in Optical Lattices: Emergence of Novel Features in Extended States. *Entropy* **2016**, *18*, 118.
https://doi.org/10.3390/e18040118

**AMA Style**

Watanabe G, Venkatesh BP, Dasgupta R.
Nonlinear Phenomena of Ultracold Atomic Gases in Optical Lattices: Emergence of Novel Features in Extended States. *Entropy*. 2016; 18(4):118.
https://doi.org/10.3390/e18040118

**Chicago/Turabian Style**

Watanabe, Gentaro, B. Prasanna Venkatesh, and Raka Dasgupta.
2016. "Nonlinear Phenomena of Ultracold Atomic Gases in Optical Lattices: Emergence of Novel Features in Extended States" *Entropy* 18, no. 4: 118.
https://doi.org/10.3390/e18040118