Bose-Hubbard Dimer Research Articles

Statistical and dynamical aspects of quantum chaos in a kicked Bose-Hubbard dimer

Statistical and dynamical aspects of quantum chaos in a kicked Bose-Hubbard dimer

The dynamics of an open Bose–Hubbard dimer with effective asymmetric coupling

The dynamics of an open Bose–Hubbard dimer with effective asymmetric coupling

Dynamical transition from localized to uniform scrambling in locally hyperbolic systems.

Fast scrambling of quantum correlations, reflected by the exponential growth of out-of-time-order correlators (OTOCs) on short pre-Ehrenfest time scales, is commonly considered as a major quantum signature of unstable dynamics in quantum systems with a classical limit. In two recent works [Phys.Rev. Lett. 123, 160401 (2019)0031-900710.1103/PhysRevLett.123.160401] and [Phys. Rev.Lett. 124, 140602 (2020)10.1103/PhysRevLett.124.140602], a significant difference in the scrambling rate of integrable (many-body) systems was observed, depending on the initial state being semiclassically localized around unstable fixed points or fully delocalized (infinite temperature). Specifically, the quantum Lyapunov exponent λ_{q} quantifying the OTOC growth is given, respectively, by λ_{q}=2λ_{s} or λ_{q}=λ_{s} in terms of the stability exponent λ_{s} of the hyperbolic fixed point. Here we show that a wave packet, initially localized around this fixed point, features a distinct dynamical transition between these two regions. We present an analytical semiclassical approach providing a physical picture of this phenomenon, and support our findings by extensive numerical simulations in the whole parameter range of locally unstable dynamics of a Bose-Hubbard dimer. Our results suggest that the existence of this crossover is a hallmark of unstable separatrix dynamics in integrable systems, thus opening the possibility to distinguish the latter, on the basis of this particular observable, from genuine chaotic dynamics generally featuring uniform exponential growth of the OTOC.

Critical optical-hysteresis regime and optical-bistability transformation

In this paper, we study the optical-hysteresis regime in a driven-dissipative Bose-Hubbard dimer under a symmetric configuration and analyze the classical optical bistability with the Gross-Pitaevskii mean-field approach. We find that the critical point of the optical-hysteresis regime can be determined by the classical optical-bistability transformation predicted with a steady state equation, and the critical tunneling rate is given by analyzing the bistable threshold points. In addition, we apply the same approach to a single nonlinear microcavity coupled with an atom and the critical atom-cavity coupling strength of the optical-hysteresis regime is also obtained. Finally, in order to illustrate that the critical point could be determined by the classical optical-bistability transformation, we analyze the Liouvillian gap and find it opens at the critical point of the optical-hysteresis regime. This work clarifies the relation between optical hysteresis and classical optical bistability, which provides theoretical references for the modulation of optical hysteresis in experiment.

Exactly solvable model behind Bose-Hubbard dimers, Ince-Gauss beams, and aberrated optical cavities

The authors present a widely applicable, solvable model (an isotropic two-dimensional harmonic oscillator subject to two simple perturbations) and show that it underlies a broad range of physical phenomena displaying a topological transition, including aberrated optical cavities that support Ince-Gauss beams as their modes and the Bose-Hubbard dimer describing two coupled superfluids.

Quantum Manifestation of the Classical Bifurcation in the Driven Dissipative Bose-Hubbard Dimer.

We analyze the classical and quantum dynamics of the driven dissipative Bose-Hubbard dimer. Under variation of the driving frequency, the classical system is shown to exhibit a bifurcation to the limit cycle, where its steady-state solution corresponds to periodic oscillation with the frequency unrelated to the driving frequency. This bifurcation is shown to lead to a peculiarity in the stationary single-particle density matrix of the quantum system. The case of the Bose-Hubbard trimer, where the discussed limit cycle bifurcates into a chaotic attractor, is briefly discussed.

Self-pulsing and chaos in the asymmetrically driven dissipative photonic Bose-Hubbard dimer: A bifurcation analysis.

We perform a systematic study of the temporal dynamics emerging in the asymmetrically driven dissipative Bose-Hubbard dimer model. This model successfully describes the nonlinear dynamics of photonic diatomic molecules in linearly coupled Kerr resonators coherently excited by a single laser beam. Such temporal dynamics may include self-pulsing oscillations, period doubled oscillatory states, chaotic dynamics, and spikes. We have thoroughly characterized such dynamical states, their origin, and their regions of stability by applying bifurcation analysis and dynamical system theory. This approach has allowed us to identify and classify the instabilities, which are responsible for the appearance of different types of temporal dynamics.

Steady-state quantum Zeno effect of driven-dissipative bosons with dynamical mean-field theory

We study a driven-dissipative Bose-Hubbard model in presence of two-particle losses and an incoherent single-particle drive on each lattice site, leading to a finite-density stationary state. Using dynamical mean-field theory (DMFT) and an impurity solver based on exact diagonalization of the associated Lindbladian, we investigate the regime of strong two-particle losses. Here, a stationary-state quantum Zeno effect emerges, as can be seen in the on-site occupation and spectral function. We show that DMFT captures this effect through its self-consistent bath. We show that, in the deep Zeno regime, the bath structure simplifies, with the occupation of all bath sites except one becoming exponentially suppressed. As a result, an effective dissipative hard-core Bose-Hubbard dimer model emerges, where the auxiliary bath site has single-particle dissipation controlled by the Zeno dissipative scale.

Dynamics of mode entanglement induced by particle-tunneling in the extended Bose–Hubbard dimer model

The evolution of mode entanglement is analyzed for a system of two indistinguishable bosons with two accessible modes. Whereas entanglement remains stationary whenever the number of bosons in each mode is left invariant, it exhibits a rich dynamics under the effects of single-and two-particle tunneling. By analyzing such effects in paradigmatic families of states, our results provide guidance for the design and control of specific dynamics of mode entanglement, by varying the tunneling transition rates and the preparation of the initial state.

Spontaneous Symmetry Breaking in a Coherently Driven Nanophotonic Bose-Hubbard Dimer.

We report on the first experimental observation of spontaneous mirror symmetry breaking (SSB) in coherently driven-dissipative coupled optical cavities. SSB is observed as the breaking of the spatial or mirror Z_{2} symmetry between two symmetrically pumped and evanescently coupled photonic crystal nanocavities, and manifests itself as random intensity localization in one of the two cavities. We show that, in a system featuring repulsive boson interactions (U>0), the observation of a pure pitchfork bifurcation requires negative photon hopping energies (J<0), which we have realized in our photonic crystal molecule. SSB is observed over a wide range of the two-dimensional parameter space of driving intensity and detuning, where we also find a region that exhibits bistable symmetric behavior. Our results pave the way for the experimental study of limit cycles and deterministic chaos arising from SSB, as well as the study of nonclassical photon correlations close to SSB transitions.

Semiclassical bifurcations and quantum trajectories: a case study of the open Bose–Hubbard dimer

We consider the open two-site Bose–Hubbard dimer, a well-known quantum mechanical model that has been realised recently for photons in two coupled photonic crystal nanocavities. The system is described by a Lindblad master equation which, for large numbers of photons, gives rise to a limiting semiclassical model in the form of a four-dimensional vector field. From the situation where both sites trap the same amount of photons under symmetric pumping, one encounters a transition that involves symmetry breaking, the creation of periodic oscillations and multistability as the pump strength is increased. We show that the associated one-parameter bifurcation diagram of the semiclassical model captures the essence of statistical properties of computed quantum trajectories as the pump strength is increased. Even for small numbers of photons, the fingerprint of the semiclassical bifurcations can be recognised reliably in observables of quantum trajectories.

Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $ \mathbb{R}^4 $

&lt;p style='text-indent:20px;'&gt;The Bose-Hubbard dimer model is a celebrated fundamental quantum mechanical model that accounts for the dynamics of bosons at two interacting sites. It has been realized experimentally by two coupled, driven and lossy photonic crystal nanocavities, which are optical devices that operate with only a few hundred photons due to their extremely small size. Our work focuses on characterizing the different dynamics that arise in the semiclassical approximation of such driven-dissipative photonic Bose-Hubbard dimers. Mathematically, this system is a four-dimensional autonomous vector field that describes two specific coupled oscillators, where both the amplitude and the phase are important. We perform a bifurcation analysis of this system to identify regions of different behavior as the pump power &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ f $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and the detuning &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \delta $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; of the driving signal are varied, for the case of fixed positive coupling. The bifurcation diagram in the &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ (f, \delta) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-plane is organized by two points of codimension-two bifurcations——a &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \mathbb{Z}_2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-equivariant homoclinic flip bifurcation and a Bykov T-point——and provides a roadmap for the observable dynamics, including different types of chaotic behavior. To illustrate the overall structure and different accumulation processes of bifurcation curves and associated regions, our bifurcation analysis is complemented by the computation of kneading invariants and of maximum Lyapunov exponents in the &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ (f, \delta) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-plane. The bifurcation diagram displays a menagerie of dynamical behavior and offers insights into the theory of global bifurcations in a four-dimensional phase space, including novel bifurcation phenomena such as degenerate singular heteroclinic cycles.&lt;/p&gt;

Self-pulsing in driven-dissipative photonic Bose-Hubbard dimers

We experimentally investigate the nonlinear dynamics of two coupled fiber ring resonators, coherently driven by a single laser beam. We comprehensively explore the optical switching arising when scanning the detuning of the undriven cavity, and show how the driven cavity detuning dramatically changes the resulting hysteresis cycle. By driving the photonic dimer out-of-equilibrium, we observe the occurrence of stable self-switching oscillations near avoided resonance crossings. All results agree well with the driven-dissipative Bose-Hubbard dimer model in the weakly coupled regime.

Repulsively diverging gradient of the density functional in the reduced density matrix functional theory

The reduced density matrix functional theory (RDMFT) is a remarkable tool for studying properties of ground states of strongly interacting quantum many body systems. As it gives access to the one-particle reduced density matrix of the ground state, it provides a perfectly tailored approach to studying the Bose–Einstein condensation (BEC) or systems of strongly correlated electrons. In particular, for homogeneous BECs as well as for the Bose–Hubbard dimer it has been recently shown that the relevant density functional exhibits a repulsive gradient (called the Bose–Einstein condensation force) which diverges when the fraction of non-condensed bosons tends to zero. In this paper, we show that the existence of the Bose–Einstein condensation force is completely universal for any type of pair-interaction and also in the non-homogeneous gases. To this end, we construct a universal family of variational trial states which allows us to suitably approximate the relevant density functional in a finite region around the set of the completely condensed states. We also show the existence of an analogous repulsive gradient in the fermionic RDMFT for the N-fermion singlet sector in the vicinity of the set of the Hartree–Fock states. Finally, we show that our approximate functional may perform well in electron transfer calculations involving low numbers of electrons. This is demonstrated numerically in the Fermi–Hubbard model in the strongly correlated limit where some other approximate functionals are known to fail.

Signatures of self-trapping in the driven-dissipative Bose–Hubbard dimer

We investigate signatures of a self-trapping transition in the driven-dissipative Bose Hubbard dimer, in presence of incoherent pump and single-particle losses. For fully symmetric couplings the stationary state density matrix is independent of any Hamiltonian parameter, and cannot therefore capture the competition between hopping-induced delocalization and the interaction-dominated self-trapping regime. We focus instead on the exact quantum dynamics of the particle imbalance after the system is prepared in a variety of initial states, and on the frequency-resolved spectral properties of the steady state, as encoded in the single-particle Green’s functions. We find clear signatures of a localization-delocalization crossover as a function of hopping to interaction ratio. We further show that a finite a pump-loss asymmetry restores a delocalization crossover in the steady-state imbalance and leads to a finite intra-dimer dissipation.

Emergence of a Renormalized 1/N Expansion in Quenched Critical Many-Body Systems.

We consider the fate of 1/N expansions in unstable many-body quantum systems, as realized by a quench across criticality, and show the emergence of e^{2λt}/N as a renormalized parameter ruling the quantum-classical transition and accounting nonperturbatively for the local divergence rate λ of mean-field solutions. In terms of e^{2λt}/N, quasiclassical expansions of paradigmatic examples of criticality, like the self-trapping transition in an integrable Bose-Hubbard dimer and the generic instability of attractive bosonic systems toward soliton formation, are pushed to arbitrarily high orders. The agreement with numerical simulations supports the general nature of our results in the appropriately combined long-time λt→∞ quasiclassical N→∞ regime, out of reach of expansions in the bare parameter 1/N. For scrambling in many-body hyperbolic systems, our results provide formal grounds to a conjectured multiexponential form of out-of-time-ordered correlators.

Saddle-point scrambling without thermalization

Out-of-time-order correlators (OTOCs) have proven to be a useful tool for studying thermalisation in quantum systems. In particular, the exponential growth of OTOCS, or scrambling, is sometimes taken as an indicator of chaos in quantum systems, despite the fact that saddle points in integrable systems can also drive rapid growth in OTOCs. By analysing the Dicke model and a driven Bose-Hubbard dimer, we demonstrate that the OTOC growth driven by chaos can, nonetheless, be distinguished from that driven by saddle points through the long-term behaviour. Besides quantitative differences in the long-term average, the saddle point gives rise to large oscillations not observed in the chaotic case. The differences are also highlighted by entanglement entropy, which in the chaotic driven dimer matches a Page curve prediction. These results illustrate additional markers that can be used to distinguish chaotic behaviour in quantum systems, beyond the initial exponential growth in OTOCs.

Spin Squeezing in a Tunneling-Dissipative Bose-Hubbard Dimer

We investigate spin squeezing of a Bose-Einstein condensate in an open symmetrical double-well trap with dissipation in the tunneling process, which can be treated as a tunneling-dissipative Bose-Hubbard dimer. Perturbation technique based on the system is applied. Its shown that more squeezing can be achieved by increasing the tunneling rate and the dissipation.

Thermalization in a Bose-Hubbard dimer with modulated tunneling

The periodically modulated Bose-Hubbard dimer model offers an experimentally realizable and highly tunable platform for observing the scrambling of quantum information and the apparent thermalisation of isolated, interacting quantum many-body systems. In this work we apply fidelity out-of-time-order correlators to establish connections between thermalisation in Floquet system, the exponential growth of FOTOCs as quantified by a non-zero quantum Lyapunov exponent, and the underlying classical transition from regular to chaotic dynamics in the dimer. Moreover, we demonstrate that a non-zero quantum Lyapunov exponent can also be inferred from measures quantifying the delocalisation of the Floquet modes of the system such as the Shannon entropy, which approaches unity if the system thermalises to the periodic Gibbs ensemble prediction.

The driven-dissipative Bose–Hubbard dimer: phase diagram and chaos

We present the phase diagram of the mean-field driven-dissipative Bose–Hubbard dimer model. For a dimer with repulsive on-site interactions (U > 0) and coherent driving, we prove that -symmetry breaking, via pitchfork bifurcations with sizable extensions of the asymmetric solutions, require a negative tunneling parameter (J < 0). In addition, we show that the model exhibits deterministic dissipative chaos. The chaotic attractor emerges from a Shilnikov mechanism of a periodic orbit born in a Hopf bifurcation and, depending on its symmetry properties, it is either localized or not.