# Hyperchaos in a Bose-Hubbard Chain with Rydberg-Dressed Interactions

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## Abstract

**:**

## 1. Introduction

## 2. Model and Method

#### 2.1. Extended Bose-Hubbard Model in the Semiclassical Limit

#### 2.2. Nonlinear Eigenenergies and Bogoliubov Spectra

#### 2.3. Poincaré Sections and Lyapunov Exponents

#### 2.4. Quenching Schemes

**Scheme$\mathbf{I}$**: First, we consider a linear quench of the potential bias [71]. The bias between two neighboring sites is given by the following function:

**Scheme$\mathbf{II}$**: Alternatively, we consider a hysteresis quench [74,75,76] where the system begins at ${\gamma}_{i}$ and then evolves to ${\gamma}_{f}$. At time $\tau ={\gamma}_{f}/\alpha $, the potential bias is quenched back towards ${\gamma}_{i}$. The function describing this scheme is as follows:

**Scheme$\mathbf{III}$**: In addition to quenching the level bias, we also change the two-body interaction strength through a linear ramp:

**Scheme$\mathbf{IV}$**: The hysteresis counterpart of the interaction quench is given by the following equation:

## 3. Stability of the Ground State

#### 3.1. Eigenenergies, Bogoliubov Spectra, and Lyapunov Exponents

#### 3.2. Quench Dynamics

## 4. Stability of the Localized State

#### 4.1. Bogoliubov Spectra and Lyapunov Exponents

#### 4.2. Quench Dynamics

## 5. Discussion

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Robert, P.S.; Zoran, H. Effects of Interactions on Bose-Einstein Condensation of an Atomic Gas. In Physics of Quantum Fluids: New Trends and Hot Topics in Atomic and Polariton Condensates; Alberto, B., Michele, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2013; pp. 341–359. [Google Scholar]
- Pethick, C.J.; Smith, H. Microscopic theory of the Bose gas. In Bose–Einstein Condensation in Dilute Gases; Cambridge University Press: Cambridge, UK, 2008. [Google Scholar]
- Leonardo, F.; Chiara, F.; Jessica, E.L.; Massimo, I. Bose-Einstein condensate in an optical lattice with tunable spacing: Transport and static properties. Opt. Express
**2005**, 13, 4303–4313. [Google Scholar] - Smerzi, A.; Fantoni, S.; Giovanazzi, S.; Shenoy, S.R. Quantum coherent atomic tunneling between two trapped bose-einstein condensates. Phys. Rev. Lett.
**1997**, 79, 4950–4953. [Google Scholar] [CrossRef] [Green Version] - Manjun, M.R.; Navarro, R.; Carretero-González, R. Solitons riding on solitons and the quantum newton’s cradle. Phys. Rev. E
**2016**, 93, 022202. [Google Scholar] - Anderson, B.P.; Haljan, P.C.; Regal, C.A.; Feder, D.L.; Collins, L.A.; Clark, C.W.; Cornell, E.A. Watching dark solitons decay into vortex rings in a bose-einstein condensate. Phys. Rev. Lett.
**2001**, 86, 2926–2929. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zachary, D.; Michael, B.; Christopher, S.; Lene Vestergaard, H. Observation of quantum shock waves created with ultra- compressed slow light pulses in a bose-einstein condensate. Science
**2001**, 293, 663–668. [Google Scholar] - Denschlag, J.; Simsarian, J.E.; Feder, D.L.; Clark, C.W.; Collins, L.A.; Cubizolles, J.; Deng, L.; Hagley, E.W.; Helmerson, K.; Phillips, W.D.; et al. Generating solitons by phase engineering of a bose-einstein condensate. Science
**2000**, 287, 97–101. [Google Scholar] [CrossRef] [PubMed] - Burger, S.; Bongs, K.; Dettmer, S.; Ertmer, W.; Sengstock, K.; Sanpera, A.; Shlyapnikov, G.V.; Lewenstein, M. Dark solitons in bose-einstein condensates. Phys. Rev. Lett.
**1999**, 83, 5198–5201. [Google Scholar] [CrossRef] [Green Version] - Cornish, S.L.; Thompson, S.T.; Wieman, C.E. Formation of bright matter-wave solitons during the collapse of attractive bose-einstein condensates. Phys. Rev. Lett.
**2006**, 96, 170401. [Google Scholar] [CrossRef] [Green Version] - Khaykovich, L.; Schreck, F.; Ferrari, G.; Bourdel, T.; Cubizolles, J.; Carr, L.D.; Castin, Y.; Salomon, C. Formation of a matter-wave bright soliton. Science
**2002**, 296, 1290–1293. [Google Scholar] [CrossRef] [Green Version] - Strecker, K.E.; Partridge, G.B.; Truscott, A.G.; Hulet, R.G. Formation and propagation of matter-wave soliton trains. Nature
**2002**, 417, 150–153. [Google Scholar] [CrossRef] - Kinoshita, T.; Wenger, T.; Weiss, D.S. A quantum newton’s cradle. Nature
**2006**, 440, 900–903. [Google Scholar] [CrossRef] - Xia, B.; Hai, W.; Chong, G. Stability and chaotic behavior of a two-component Bose-Einstein condensate. Phys. Lett. Sect. A Gen. At. Solid State Phys.
**2006**, 351, 136–142. [Google Scholar] [CrossRef] - Liu, B.; Fu, L.B.; Yang, S.P.; Liu, J. Josephson oscillation and transition to self-trapping for Bose-Einstein condensates in a triple-well trap. Phys. Rev. A
**2007**, 75, 033601. [Google Scholar] [CrossRef] [Green Version] - Graefe, E.M.; Korsch, H.J.; Witthaut, D. Mean-field dynamics of a Bose-Einstein condensate in a time-dependent triple-well trap: Nonlinear eigenstates, Landau-Zener models, and stimulated Raman adiabatic passage. Phys. Rev. A
**2006**, 73, 013617. [Google Scholar] [CrossRef] [Green Version] - Viscondi, T.F.; Furuya, K. Dynamics of a Bose-Einstein condensate in a symmetric triple-well trap. J. Phys. A Math. Theor.
**2011**, 44, 175301. [Google Scholar] [CrossRef] [Green Version] - Li, L.; Wang, B.; Lü, X.Y.; Wu, Y. Chaos-related Localization in Modulated Lattice Array. Annalen der Physik
**2018**, 530, 1700218. [Google Scholar] [CrossRef] - Chong, G.; Hai, W.; Xie, Q. Controlling chaos in a weakly coupled array of Bose-Einstein condensates. Phys. Rev. E
**2005**, 71, 016202. [Google Scholar] [CrossRef] - Liu, J.; Fu, L.; Ou, B.Y.; Chen, S.G.; Choi, D.I.; Wu, B.; Niu, Q. Theory of nonlinear Landau-Zener tunneling. Phys. Rev. A
**2002**, 66, 023404. [Google Scholar] [CrossRef] [Green Version] - Liu, J.; Wu, B.; Niu, Q. Nonlinear Evolution of Quantum States in the Adiabatic Regime. Phys. Rev. Lett.
**2003**, 90, 170404. [Google Scholar] [CrossRef] [Green Version] - Albiez, M.; Gati, R.; Fölling, J.; Hunsmann, S.; Cristiani, M.; Oberthaler, M.K. Direct observation of tunneling and nonlinear self-trapping in a single bosonic josephson junction. Phys. Rev. Lett.
**2005**, 95, 010402. [Google Scholar] [CrossRef] [Green Version] - Zibold, T.; Nicklas, E.; Gross, C.; Oberthaler, M.K. Classical bifurcation at the transition from rabi to Josephson dynamics. Phys. Rev. Lett.
**2010**, 105, 204101. [Google Scholar] [CrossRef] - Gotlibovych, I.; Schmidutz, T.F.; Gaunt, A.L.; Navon, N.; Smith, R.P.; Hadzibabic, Z. Observing properties of an interacting homogeneous bose-einstein condensate: Heisenberg-limited momentum spread, interaction energy, and free-expansion dynamics. Phys. Rev. A
**2014**, 89, 061604. [Google Scholar] [CrossRef] [Green Version] - Gaunt, A.L.; Schmidutz, T.F.; Gotlibovych, I.; Smith, R.P.; Hadzibabic, Z. Bose-einstein condensation of atoms in a uniform potential. Phys. Rev. Lett.
**2013**, 110, 200406. [Google Scholar] [CrossRef] [PubMed] - Schmidutz, T.F.; Gotlibovych, I.; Gaunt, A.L.; Smith, R.P.; Navon, N.; Hadzibabic, Z. Quantum joule-thomson effect in a saturated homogeneous bose gas. Phys. Rev. Lett.
**2014**, 112, 040403. [Google Scholar] [CrossRef] [Green Version] - Buonsante, P.; Penna, V. Some remarks on the coherent-state variational approach to nonlinear boson models. J. Phys. A Math. Theor.
**2008**, 41, 175301. [Google Scholar] [CrossRef] - Hai, W.; Rong, S.; Zhu, Q. Discrete chaotic states of a Bose-Einstein condensate. Phys. Rev. E
**2008**, 78, 066214. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sinha, S.; Sinha, S. Chaos and Quantum Scars in Bose-Josephson Junction Coupled to a Bosonic Mode. Phys. Rev. Lett.
**2020**, 125, 134101. [Google Scholar] [CrossRef] [PubMed] - Boukobza, E.; Moore, M.G.; Cohen, D.; Vardi, A. Nonlinear phase dynamics in a driven bosonic josephson junction. Phys. Rev. Lett.
**2010**, 104, 240402. [Google Scholar] [CrossRef] [Green Version] - Pedri, P.; Santos, L. Two-Dimensional Bright Solitons in Dipolar Bose-Einstein Condensates. Phys. Rev. Lett.
**2005**, 95, 200404. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tikhonenkov, I.; Malomed, B.A.; Vardi, A. Anisotropic Solitons in Dipolar Bose-Einstein Condensates. Phys. Rev. Lett.
**2008**, 100, 090406. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Nath, R.; Pedri, P.; Santos, L. Phonon Instability with Respect to Soliton Formation in Two-Dimensional Dipolar Bose-Einstein Condensates. Phys. Rev. Lett.
**2009**, 102, 050401. [Google Scholar] [CrossRef] [Green Version] - Cuevas, J.; Malomed, B.A.; Kevrekidis, P.G.; Frantzeskakis, D.J. Solitons in quasi-one-dimensional Bose-Einstein condensates with competing dipolar and local interactions. Phys. Rev. A
**2009**, 79, 053608. [Google Scholar] [CrossRef] [Green Version] - Young-S, L.E.; Muruganandam, P.; Adhikari, S.K. Dynamics of quasi-one-dimensional bright and vortex solitons of a dipolar Bose–Einstein condensate with repulsive atomic interaction. J. Phys. B At. Mol. Opt. Phys.
**2011**, 44, 101001. [Google Scholar] [CrossRef] [Green Version] - Lahaye, T.; Pfau, T.; Santos, L. Mesoscopic ensembles of polar bosons in triple-well potentials. Phys. Rev. Lett.
**2010**, 104, 170404. [Google Scholar] [CrossRef] [Green Version] - Xiong, B.; Fischer, U.R. Interaction-induced coherence among polar bosons stored in triple-well potentials. Phys. Rev. A
**2013**, 88, 063608. [Google Scholar] [CrossRef] [Green Version] - Gallemí, A.; Guilleumas, M.; Mayol, R.; Sanpera, A. Role of anisotropy in dipolar bosons in triple-well potentials. Phys. Rev. A
**2013**, 88, 063645. [Google Scholar] [CrossRef] [Green Version] - Köberle, P.; Cartarius, H.; Fabčič, T.; Main, J.; Wunner, G. Bifurcations, order and chaos in the bose–einstein condensation of dipolar gases. New J. Phys.
**2009**, 11, 023017. [Google Scholar] [CrossRef] - Andreev, P.A. Quantum hydrodynamic theory of quantum fluctuations in dipolar bose–einstein condensate. Chaos An Interdiscip. J. Nonlinear Sci.
**2021**, 31, 023120. [Google Scholar] [CrossRef] - Xiong, B.; Gong, J.; Pu, H.; Bao, W.; Li, B. Symmetry breaking and self-trapping of a dipolar Bose-Einstein condensate in a double-well potential. Phys. Rev. A
**2009**, 79, 013626. [Google Scholar] [CrossRef] [Green Version] - Abad, M.; Guilleumas, M.; Mayol, R.; Pi, M.; Jezek, D.M. A dipolar self-induced bosonic Josephson junction. Europhys. Lett.
**2011**, 94, 10004. [Google Scholar] [CrossRef] [Green Version] - Wang, C.; Kevrekidis, P.G.; Frantzeskakis, D.J.; Malomed, B.A. Effects of long-range nonlinear interactions in double-well potentials. Phys. D Nonlinear Phenom.
**2011**, 240, 805–813. [Google Scholar] [CrossRef] [Green Version] - Adhikari, S.K. Self-trapping of a dipolar Bose-Einstein condensate in a double well. Phys. Rev. A
**2014**, 89, 043609. [Google Scholar] [CrossRef] [Green Version] - Zhang, A.-X.; Xue, J.-K. Dipolar-induced interplay between inter-level physics and macroscopic phase transitions in triple-well potentials. J. Phys. B At. Mol. Opt. Phys.
**2012**, 45, 145305. [Google Scholar] [CrossRef] - Fortanier, R.; Zajec, D.; Main, J.; Wunner, G. Dipolar Bose–Einstein condensates in triple-well potentials. J. Phys. B At. Mol. Opt. Phys.
**2013**, 46, 235301. [Google Scholar] [CrossRef] [Green Version] - Bouchoule, I.; Molmer, K. Spin squeezing of atoms by the dipole interaction in virtually excited Rydberg states. Phys. Rev. A
**2002**, 65, 041803. [Google Scholar] [CrossRef] - Henkel, N.; Nath, R.; Pohl, T. Three-dimensional roton excitations and supersolid formation in rydberg-excited bose-einstein condensates. Phys. Rev. Lett.
**2010**, 104, 195302. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Honer, J.; Weimer, H.; Pfau, T.; Büchler, H.P. Collective many-body interaction in rydberg dressed atoms. Phys. Rev. Lett.
**2010**, 105, 160404. [Google Scholar] [CrossRef] [Green Version] - Pupillo, G.; Micheli, A.; Boninsegni, M.; Lesanovsky, I.; Zoller, P. Strongly correlated gases of rydberg-dressed atoms: Quantum and classical dynamics. Phys. Rev. Lett.
**2010**, 104, 223002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Johnson, J.E.; Rolston, S.L. Interactions between Rydberg-dressed atoms. Phys. Rev. A
**2010**, 82, 033412. [Google Scholar] [CrossRef] [Green Version] - Li, W.; Hamadeh, L.; Lesanovsky, I. Probing the interaction between Rydberg-dressed atoms through interference. Phys. Rev. A
**2012**, 85, 053615. [Google Scholar] [CrossRef] [Green Version] - DeSalvo, B.J.; Aman, J.A.; Gaul, C.; Pohl, T.; Yoshida, S.; Burgdörfer, J.; Hazzard, K.R.A.; Dunning, F.B.; Killian, T.C. Rydberg-blockade effects in autler-townes spectra of ultracold strontium. Phys. Rev. A
**2016**, 93, 022709. [Google Scholar] [CrossRef] [Green Version] - Hsueh, C.-H.; Wang, C.-W.; Wu, W.-C. Vortex structures in a rotating rydberg-dressed bose-einstein condensate with the lee-huang-yang correction. Phys. Rev. A
**2020**, 102, 063307. [Google Scholar] [CrossRef] - Maucher, F.; Henkel, N.; Saffman, M.; Królikowski, W.; Skupin, S.; Pohl, T. Rydberg-induced solitons: Three-dimensional self-trapping of matter waves. Phys. Rev. Lett.
**2011**, 106, 170401. [Google Scholar] [CrossRef] [PubMed] - Cinti, F.; MacRì, T.; Lechner, W.; Pupillo, G.; Pohl, T. Defect-induced supersolidity with soft-core bosons. Nat. Commun.
**2014**, 5, 4235. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hsueh, C.H.; Tsai, Y.C.; Wu, W.C. Excitations of one-dimensional supersolids with optical lattices. Phys. Rev. A
**2016**, 93, 063605. [Google Scholar] [CrossRef] - McCormack, G.; Nath, R.; Li, W. Dynamical excitation of maxon and roton modes in a Rydberg-Dressed Bose-Einstein Condensate. Phys. Rev. A
**2020**, 102, 023319. [Google Scholar] [CrossRef] - Lauer, A.; Muth, D.; Fleischhauer, M. Transport-induced melting of crystals of Rydberg dressed atoms in a one-dimensional lattice. New J. Phys.
**2012**, 14, 095009. [Google Scholar] [CrossRef] [Green Version] - Lan, Z.; Minar, J.; Levi, E.; Li, W.; Lesanovsky, I. Emergent Devil’s Staircase without Particle-Hole Symmetry in Rydberg Quantum Gases with Competing Attractive and Repulsive Interactions. Phys. Rev. Lett.
**2015**, 115, 203001. [Google Scholar] [CrossRef] [Green Version] - Angelone, A.; Mezzacapo, F.; Pupillo, G. Superglass Phase of Interaction-Blockaded Gases on a Triangular Lattice. Phys. Rev. Lett.
**2016**, 116, 135303. [Google Scholar] [CrossRef] [Green Version] - Chougale, Y.; Nath, R. Ab initio calculation of Hubbard parameters for Rydberg-dressed atoms in a one-dimensional optical lattice. J. Phys. B At. Mol. Opt. Phys.
**2016**, 49, 144005. [Google Scholar] [CrossRef] - Li, Y.; Geißler, A.; Hofstetter, W.; Li, W. Supersolidity of lattice bosons immersed in strongly correlated Rydberg dressed atoms. Phys. Rev. A
**2018**, 97, 023619. [Google Scholar] [CrossRef] [Green Version] - Zhou, Y.; Li, Y.; Nath, R.; Li, W. Quench dynamics of Rydberg-dressed bosons on two-dimensional square lattices. Phys. Rev. A
**2020**, 101, 013427. [Google Scholar] [CrossRef] [Green Version] - Barbier, M.; Geißler, A.; Hofstetter, W. Decay-dephasing-induced steady states in bosonic rydberg-excited quantum gases in an optical lattice. Phys. Rev. A
**2019**, 99, 033602. [Google Scholar] [CrossRef] [Green Version] - Jau, Y.Y.; Hankin, A.M.; Keating, T.; Deutsch, I.H.; Biedermann, G.W. Entangling atomic spins with a Rydberg-dressed spin-flip blockade. Nat. Phys.
**2016**, 12, 3487. [Google Scholar] [CrossRef] [Green Version] - Zeiher, J.; van Bijnen, R.; Schauß, P.; Hild, S.; Choi, J.Y.; Pohl, T.; Bloch, I.; Gross, C. Many-body interferometry of a Rydberg-dressed spin lattice. Nat. Phys.
**2016**, 12, 3835. [Google Scholar] [CrossRef] - Zeiher, J.; Choi, J.Y.; Rubio-Abadal, A.; Pohl, T.; Van Bijnen, R.; Bloch, I.; Gross, C. Coherent many-body spin dynamics in a long-range interacting Ising chain. Phys. Rev. X
**2017**, 7, 041063. [Google Scholar] [CrossRef] [Green Version] - Guardado-Sanchez, E.; Spar, B.M.; Schauss, P.; Belyansky, R.; Young, J.T.; Bienias, P.; Gorshkov, A.V.; Iadecola, T.; Bakr, W.S. Quench Dynamics of a Fermi Gas with Strong Long-Range Interactions. arXiv
**2020**, arXiv:2010.05871. [Google Scholar] - Borish, V.; Marković, O.; Hines, J.A.; Rajagopal, S.V.; Schleier-Smith, M. Transverse-Field Ising Dynamics in a Rydberg-Dressed Atomic Gas. Phys. Rev. Lett.
**2020**, 124, 063601. [Google Scholar] [CrossRef] [Green Version] - McCormack, G.; Nath, R.; Li, W. Nonlinear dynamics of Rydberg-dressed Bose-Einstein condensates in a triple-well potential. Phys. Rev. A
**2020**, 102, 063329. [Google Scholar] [CrossRef] - Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenom.
**1985**, 16, 285–317. [Google Scholar] [CrossRef] [Green Version] - Andreev, A.V.; Balanov, A.G.; Fromhold, T.M.; Greenaway, M.T.; Hramov, A.E.; Li, W.; Makarov, V.V.; Zagoskin, A.M. Emergence and control of complex behaviors in driven systems of interacting qubits with dissipation. NPJ Quantum Inf.
**2021**, 7, 1–7. [Google Scholar] [CrossRef] - Eckel, S.; Lee, J.G.; Jendrzejewski, F.; Murray, N.; Clark, C.W.; Lobb, C.J.; Phillips, W.D.; Edwards, M.; Campbell, G.K. Hysteresis in a quantized superfluid ’atomtronic’ circuit. Nature
**2014**, 506, 200–203. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Trenkwalder, A.; Spagnolli, G.; Semeghini, G.; Coop, S.; Landini, M.; Castilho, P.; Pezzè, L.; Modugno, G.; Inguscio, M.; Smerzi, A.; et al. Quantum phase transitions with parity-symmetry breaking and hysteresis. Nat. Phys.
**2016**, 12, 826–829. [Google Scholar] [CrossRef] [Green Version] - Bürkle, R.; Vardi, A.; Cohen, D.; Anglin, J.R. Probabilistic hysteresis in integrable and chaotic isolated hamiltonian systems. Phys. Rev. Lett.
**2019**, 123, 114101. [Google Scholar] [CrossRef] [Green Version] - Scully, M.O.; Zubairy, M.S. Quantum Optics, 1st ed.; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Castro, E.R.; Chávez-Carlos, J.; Roditi, I.; Santos, L.F.; Hirsch, J.G. Quantum-classical correspondence of a system of interacting bosons in a triple-well potential. arXiv
**2021**, arXiv:2105.10515. [Google Scholar] [CrossRef] - Dey, A.; Cohen, D.; Vardi, A. Adiabatic Passage through Chaos. Phys. Rev. Lett.
**2018**, 121, 250405. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dey, A.; Cohen, D.; Vardi, A. Many-body adiabatic passage: Quantum detours around chaos. Phys. Rev. A
**2019**, 99, 033623. [Google Scholar] [CrossRef] [Green Version] - Datseris, G. Dynamicalsystems.jl: A julia software library for chaos and nonlinear dynamics. J. Open Source Softw.
**2018**, 3, 598. [Google Scholar] [CrossRef] - Baier, G.; Klein, M. Maximum hyperchaos in generalized Hénon maps. Phys. Lett. A
**1990**, 151, 281–284. [Google Scholar] [CrossRef] - Baier, G.; Sahle, S. Design of hyperchaotic flows. Phys. Rev. E
**1995**, 51, R2712–R2714. [Google Scholar] [CrossRef] - Kapitaniak, T.; Thylwe, K.-E.; Cohen, I.; Wojewoda, J. Chaos-hyperchaos transition. Chaos Solitons Fractals
**1995**, 5, 2003–2011. [Google Scholar] [CrossRef] - Tarkhov, A.E.; Wimberger, S.; Fine, B.V. Extracting lyapunov exponents from the echo dynamics of bose-einstein condensates on a lattice. Phys. Rev. A
**2017**, 96, 023624. [Google Scholar] [CrossRef] [Green Version] - Lerose, A.; Pappalardi, S. Bridging entanglement dynamics and chaos in semiclassical systems. Phys. Rev. A
**2020**, 102, 032404. [Google Scholar] [CrossRef] - Pausch, L.; Carnio, E.G.; Rodríguez, A.; Buchleitner, A. Chaos and Ergodicity across the Energy Spectrum of Interacting Bosons. Phys. Rev. Lett.
**2021**, 126, 150601. [Google Scholar] [CrossRef] - Kollath, C.; Roux, G.; Biroli, G.; Läuchli, A.M. Statistical properties of the spectrum of the extended Bose-Hubbard model. J. Stat. Mech.
**2010**, 2010, P08011. [Google Scholar] [CrossRef] - Chen, Y.; Cai, Z. Persistent oscillations versus thermalization in the quench dynamics of quantum gases with long-range interactions. Phys. Rev. A
**2020**, 101, 023611. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(Color online)

**The extended Bose-Hubbard Chain and quenching schemes.**(

**a**) Nearest-neighbor (U) and next-nearest-neighbor (V) interactions between atoms in a one-dimensional optical lattice (lattice constant d). The tilting of the lattice is denoted by the parameter $\gamma $. We consider a linear quench in (

**b**) $\gamma $ and (

**c**) U towards a non-zero value (solid). When $\gamma $ (U) returns to the initial value (both solid and dashed), this is a hysteresis quench. The rate to quench $\gamma $ (U) is $\alpha $ ($\beta $). See text for details of the soft-core interaction and quenching protocols.

**Figure 2.**

**Eigenenergies, Bogoliubov spectra, and Lyapunov exponents when varying the tilt $\gamma $.**We show (

**a**) the nonlinear eigenenergy, the Bogoliubov spectra of (

**b**) the ground state and (

**c**) the first excited state, and (

**d**) the Lyapunov exponents of the ground state. The nonlinearity dominates when $\left|\gamma \right|$ is small, leading to loops in the eigenenergy. The Bogoliubov spectra are all real when the system is in the ground state (

**b**). The Bogoliubov spectra have complex components (red region) when the system is in the first-excited state. Positive Lyapunov exponents indicate the system exhibits chaos dynamically, which appear mostly in the loop region of the eigenenergy. Parameters are $L=3$ and $U=2V=5$.

**Figure 3.**

**Eigenenergies and Lyapunov exponents as a function of U.**We show eigenenergy for (

**a**) $L=3$ and (

**b**) $L=5$ when the trap is balanced ($\gamma =0$). Level crossings are found when the interaction is strong. Starting from the ground state, we calculate Lyapunov exponents for (

**c**) $L=3$ and (

**d**) $L=5$. For a given U, Lyapunov exponents of same value but opposite signs appear in pairs.

**Figure 4.**(Color online)

**Final population distribution of the ground state**. The population by quenching $\gamma $ with (

**a**) scheme $\mathbf{I}$ and (

**b**) scheme $\mathbf{II}$ is shown for $L=3$. In the numerical simulation, ${\gamma}_{i}=-{\gamma}_{f}=-10$ and the interaction strength is $U=5$. The interaction U is quenched with (

**c**) scheme $\mathbf{III}$ and (

**d**) scheme $\mathbf{IV}$, where ${U}_{i}=0$, ${U}_{f}=10$ and $\gamma =0$, respectively. In all the figures, the quench rates ($\alpha $ or $\beta $) are 1 (blue), $0.1$ (green), and $0.01$ (red). The total number of trajectories is $M=100$. The target final state is shown as the large black circle.

**Figure 5.**(Color online)

**Bogoliubov spectra and Lyapunov exponents of the localized state.**Dynamically unstable regions (dark red) for (

**a**) $L=3$ and (

**b**) L = 5 are shown as a function of U and $\gamma $. Panels (

**c**,

**d**) give the Lyapunov exponents as a function of U. Random perturbation to the initial state are examined for (

**e**) $L=3$ and (

**f**) $L=5$. The red lines show the maximal Lyapunov exponents in (

**c**,

**d**), correspondingly. Here, $\gamma =0$ in panels (

**c**,

**d**).

**Figure 6.**(Color online)

**Final population distribution of the localized state**. The first and second row show the linear and hysteresis quench of $\gamma $. Here, $U=5$, ${\gamma}_{i}=-{\gamma}_{f}=10$. The third and fourth row show the linear and hysteresis quench of U, with ${U}_{i}=0$, ${U}_{f}=10$, and $\gamma =0$. In (

**a**–

**d**), we consider three sites and the initial thermal state is $\overline{\Psi}=[0.1{e}^{i{\varphi}_{1}},\sqrt{0.98}{e}^{i{\varphi}_{2}},0.1{e}^{{\varphi}_{3}}]$, with ${\varphi}_{j}$ ($j=1,\phantom{\rule{0.166667em}{0ex}}2,\phantom{\rule{0.166667em}{0ex}}3$) being a random number in $[0,2\pi ]$. In (

**e**–

**h**), $L=5$ and the initial state is $\overline{\Psi}=[\sqrt{0.005}{e}^{i{\varphi}_{1}},\sqrt{0.98}{e}^{i{\varphi}_{2}},\sqrt{0.005}{e}^{i{\varphi}_{3}},\sqrt{0.005}{e}^{i{\varphi}_{4}},{\sqrt{0.005}}^{i{\varphi}_{5}}]$, with ${\varphi}_{j}$ ($j=1,\cdots ,5$) being randomly distributed in $[0,2\pi ]$. The small fraction in sites other than the localized state is used to trigger the hopping dynamics. The other parameters are the same as the ones in Figure 4.

**Figure 7.**(Color online)

**Lyapunov Exponents vs. System Size.**The maximum Lyapunov exponent and total number of positive Lyapunov exponents are shown in (

**a**,

**b**) for the ground state configuration. Panels (

**c**,

**d**) show the same quantity for the localized state. The larger the value of U is, the larger the maximal Lyapunov exponent. The maximal Lyapunov exponent decreases with increasing L. The number of Lyapunov exponents increases and then decreases with increasing L. For the localized state, $\eta $ increases almost linearly with increasing L. In each panel, $U=1$ (square), 3 (circle), and 5 (triangle).

**Figure 8.**(Color online)

**Poincaré Sections of the ground state and localized state on the ${\mathcal{U}}_{1}$-plane**. The Poincaré sections are shown for the ground state (

**a**–

**c**) and the localized state (

**d**–

**f**). Each point represents a numerical realization. We consider $L=5$ (

**a**,

**d**), $L=10$ (

**b**,

**e**), and $L=20$ (

**c**,

**f**). Other parameters are $U=3$ and $\gamma =0$.

**Figure 9.**(Color online)

**Areas of the Poincaré Sections**. We compare the fitted area (open shapes) of the Poincaré section with ${\lambda}_{m}$ (solid) for both the ground state (

**a**) and the localized state (

**b**), respectively. The blue circles are for $U=3$, and red triangles for $U=5$. In both situations, $\gamma =0$.

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**MDPI and ACS Style**

McCormack, G.; Nath, R.; Li, W.
Hyperchaos in a Bose-Hubbard Chain with Rydberg-Dressed Interactions. *Photonics* **2021**, *8*, 554.
https://doi.org/10.3390/photonics8120554

**AMA Style**

McCormack G, Nath R, Li W.
Hyperchaos in a Bose-Hubbard Chain with Rydberg-Dressed Interactions. *Photonics*. 2021; 8(12):554.
https://doi.org/10.3390/photonics8120554

**Chicago/Turabian Style**

McCormack, Gary, Rejish Nath, and Weibin Li.
2021. "Hyperchaos in a Bose-Hubbard Chain with Rydberg-Dressed Interactions" *Photonics* 8, no. 12: 554.
https://doi.org/10.3390/photonics8120554