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arXiv:2409.04311 · Physical Review A

Many-body tunneling in a double-well potential

M. Zendra, F. Borgonovi, G. L. Celardo, S. A. Gurvitz

Published Dec 27, 2024• 2 citations
arXivHTMLPDF

Abstract

We present an approach for evaluating Wannier functions, offering an alternative perspective on their role in many-body systems. Unlike traditional methods, such as the maximally localized Wannier functions approach, which focuses on minimizing the function tails, our approach emphasizes these tails. Using perturbative analytical approximations and extensive numerical simulations on an exactly solvable model, we address nonstandard Hubbard terms and demonstrate their critical influence on many-body dynamics. Specifically, we study tunneling dynamics in arbitrary double-well potentials, moving beyond the standard Hubbard model to include nonstandard terms such as density-induced tunneling and pair tunneling. Our results reveal that these terms significantly modify the dynamics predicted by the standard Hubbard model: Density-induced tunneling modifies the single-particle tunneling parameter ${\mathrm{\ensuremath{\Omega}}}_{0}$, while pair tunneling enables coherent propagation not captured by the standard model. We show that the discrepancies between the standard and nonstandard Hubbard models grow with increasing interaction strength, potentially leading to novel transport behaviors. However, at lower interaction strengths, both models converge, as nonstandard terms become negligible. These findings have important implications for phenomena such as superconductivity in twisted bilayer graphene and metal-insulator transitions. Our model aligns well with numerical simulations of lowest-band parameters and is strongly supported by experimental observations of second-order atom tunneling in optical double-well potentials. This strong agreement with experimental data highlights the accuracy and potential of our approach in providing a more comprehensive framework for describing complex many-body systems than the standard Hubbard model.

DOI
10.1103/physreva.110.062222

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A variationally optimised basis allows an accurate description of the quantum \nbehaviour of ultra-cold atoms, even in the strongly correlated regime. A rescaling \nscheme corrects discrepancies caused by using a reduced Hilbert space. This \napproach also allows the modelling of experimentally realizable double-well potentials, \nwhich still reveals the maximally-entangled states seen in fixed basis models. \nTime dynamics of these double-well systems show macroscopic tunnelling between \nwells for bosons with a sufficient interaction strength. \nThe many-body problem of interacting bosons in the highly-correlated regime \nis difficult. The number of basis states needed to describe this quantum system \naccurately quickly grows beyond computational reach. Rescaling the interaction \nstrength proves a simple and effective method of calculating exact eigenvalues in a \nreduced Hilbert space. \nBosonic systems in the double-well potential are investigated next. First, how \ndifferent eigen-states depend on the interaction strength is examined. The variationally \noptimised method has advantages over a standard fixed basis method with \nthe ability to model experimentally viable systems and explore more stronglycorrelated \nregimes. Secondly, tunnelling dynamics in the double well are studied, \nspecifically for a system where all particles initially occupy a single well. Oscillations \ncorresponding to collective tunnelling between wells are found in regimes \nwhere there are zero interactions or bosons lie in a maximally-entangled state. \nWhat governs the dynamics outside these two regimes is also considered.

•Jan 1, 2010•0 citations
We present an approach for evaluating Wannier functions, offering an alternative perspective on their role in many-body systems. Unlike traditional methods, such as the maximally localized Wannier functions approach, which focuses on minimizing the function tails, our approach emphasizes these tails. Using perturbative analytical approximations and extensive numerical simulations on an exactly solvable model, we address nonstandard Hubbard terms and demonstrate their critical influence on many-body dynamics. Specifically, we study tunneling dynamics in arbitrary double-well potentials, moving beyond the standard Hubbard model to include nonstandard terms such as density-induced tunneling and pair tunneling. Our results reveal that these terms significantly modify the dynamics predicted by the standard Hubbard model: Density-induced tunneling modifies the single-particle tunneling parameter \ensuremathΩ0{\mathrm{\ensuremath{\Omega}}}_{0}\ensuremathΩ0​, while pair tunneling enables coherent propagation not captured by the standard model. We show that the discrepancies between the standard and nonstandard Hubbard models grow with increasing interaction strength, potentially leading to novel transport behaviors. However, at lower interaction strengths, both models converge, as nonstandard terms become negligible. These findings have important implications for phenomena such as superconductivity in twisted bilayer graphene and metal-insulator transitions. Our model aligns well with numerical simulations of lowest-band parameters and is strongly supported by experimental observations of second-order atom tunneling in optical double-well potentials. This strong agreement with experimental data highlights the accuracy and potential of our approach in providing a more comprehensive framework for describing complex many-body systems than the standard Hubbard model.
We present a nonstandard Hubbard model applicable to arbitrary single-particle potential profiles and interparticle interactions. Our approach involves a treatment of Wannier functions, free from the ambiguities of conventional methods and applicable to finite systems without periodicity constraints. To ensure the consistent evaluation of Wannier functions, we develop a perturbative approach, utilizing the barrier penetration coefficient as a perturbation parameter. With the defined Wannier functions as a basis, we derive the Hubbard Hamiltonian, revealing the emergence of density-induced and pair tunneling terms alongside standard contributions. Our investigation demonstrates that long-range interparticle interactions can induce a mechanism for repulsive particle pairing. This mechanism relies on the effective suppression of single-particle tunneling due to density-induced tunneling. Contrary to expectations based on the standard Hubbard model, an increase in interparticle interaction does not lead to an insulating state. Instead, our proposed mechanism implies the coherent motion of correlated electron pairs, similar to bound states within a multiwell system, resistant to decay from single-electron tunneling transitions. These findings carry significant implications for various phenomena, including the formation of flat bands, the emergence of superconductivity in twisted bilayer graphene, and the possibility of a metal-insulator transition.
Originally, the Hubbard model was derived for describing the behavior of strongly correlated electrons in solids. However, for over a decade now, variations of it have also routinely been implemented with ultracold atoms in optical lattices, allowing their study in a clean, essentially defect-free environment. Here, we review some of the vast literature on this subject, with a focus on more recent non-standard forms of the Hubbard model. After giving an introduction to standard (fermionic and bosonic) Hubbard models, we discuss briefly common models for mixtures, as well as the so-called extended Bose-Hubbard models, that include interactions between neighboring sites, next-neighbor sites, and so on. The main part of the review discusses the importance of additional terms appearing when refining the tight-binding approximation for the original physical Hamiltonian. Even when restricting the models to the lowest Bloch band is justified, the standard approach neglects the density-induced tunneling (which has the same origin as the usual on-site interaction). The importance of these contributions is discussed for both contact and dipolar interactions. For sufficiently strong interactions, the effects related to higher Bloch bands also become important even for deep optical lattices. Different approaches that aim at incorporating these effects, mainly via dressing the basis, Wannier functions with interactions, leading to effective, density-dependent Hubbard-type models, are reviewed. We discuss also examples of Hubbard-like models that explicitly involve higher p orbitals, as well as models that dynamically couple spin and orbital degrees of freedom. Finally, we review mean-field nonlinear Schrödinger models of the Salerno type that share with the non-standard Hubbard models nonlinear coupling between the adjacent sites. In that part, discrete solitons are the main subject of consideration. We conclude by listing some open problems, to be addressed in the future.
We show that multi-orbital and density-induced tunneling have a significant impact on the phase diagram of bosonic atoms in optical lattices. Off-site interactions lead to density-induced hopping, the so-called bond-charge interactions, which can be identified with an effective tunneling potential and can reach the same order of magnitude as conventional tunneling. In addition, interaction-induced higher-band processes also give rise to strongly modified tunneling, on-site and bond-charge interactions. We derive an extended occupation-dependent Hubbard model with multi-orbitally renormalized processes and compute the corresponding phase diagram. It substantially deviates from the single-band Bose–Hubbard model and predicts strong changes of the superfluid-to-Mott-insulator transition. In general, the presented beyond-Hubbard physics plays an essential role in bosonic lattice systems and has an observable influence on experiments with tunable interactions.
We construct the effective lowest-band Bose-Hubbard model incorporating interaction-induced on-site correlations. The model is based on ladder operators for local correlated states, which deviate from the usual Wannier creation and annihilation, allowing for a systematic construction of the most appropriate single-band low-energy description in the form of the extended Bose-Hubbard model. A formulation of this model in terms of ladder operators not only naturally contains the previously found effective multibody interactions, but also contains multibody-induced single-particle tunneling, pair tunneling, and nearest-neighbor interaction processes of higher orders. An alternative description of the same model can be formulated in terms of occupation-dependent Bose-Hubbard parameters. These multiparticle effects can be enhanced using Feshbach resonances, leading to corrections which are well within experimental reach and of significance to the phase diagram of ultracold bosonic atoms in an optical lattice. We analyze the energy-reduction mechanism of interacting atoms on a local lattice site and show that this cannot be explained only by a spatial broadening of Wannier orbitals on a single-particle level, which neglects correlations.
We propose a Hamiltonian of ultracold spinless atom in optical lattices including the two-body interaction of nearest neighbors, which reduces to the Bose-Hubbard model in weak-interaction limit. An atom-pair hopping term appearing in the Hamiltonian explains naturally the recent experimental observation of correlated tunneling in a double-well trap with strong atom-atom interactions and moreover leads to a dynamic process of atom-pair tunneling where strongly interacting atoms can tunnel back and forth as a fragmented pair. Finally a dynamics of oscillations induced by the atom-pair tunneling is found in the strong interaction regime, where the Bose-Hubbard model gives rise to the insulator state with fixed time-averaged value of atom-occupation number only. Quantum phase transitions between two quantum phases characterized by degenerate and nondegenerate ground states are shown to be coinciding with the Landau second-order phase-transition theory. In the system of finite atom number the degeneracy of ground states can be removed by quantum tunneling for the even number of atoms but not for the odd number.
A variationally optimised basis allows an accurate description of the quantum \nbehaviour of ultra-cold atoms, even in the strongly correlated regime. A rescaling \nscheme corrects discrepancies caused by using a reduced Hilbert space. This \napproach also allows the modelling of experimentally realizable double-well potentials, \nwhich still reveals the maximally-entangled states seen in fixed basis models. \nTime dynamics of these double-well systems show macroscopic tunnelling between \nwells for bosons with a sufficient interaction strength. \nThe many-body problem of interacting bosons in the highly-correlated regime \nis difficult. The number of basis states needed to describe this quantum system \naccurately quickly grows beyond computational reach. Rescaling the interaction \nstrength proves a simple and effective method of calculating exact eigenvalues in a \nreduced Hilbert space. \nBosonic systems in the double-well potential are investigated next. First, how \ndifferent eigen-states depend on the interaction strength is examined. The variationally \noptimised method has advantages over a standard fixed basis method with \nthe ability to model experimentally viable systems and explore more stronglycorrelated \nregimes. Secondly, tunnelling dynamics in the double well are studied, \nspecifically for a system where all particles initially occupy a single well. Oscillations \ncorresponding to collective tunnelling between wells are found in regimes \nwhere there are zero interactions or bosons lie in a maximally-entangled state. \nWhat governs the dynamics outside these two regimes is also considered.