Tomasz Sowiński

Non-standard Hubbard models in optical lattices: a review

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.

Dipolar Molecules in Optical Lattices

We study the extended Bose-Hubbard model describing an ultracold gas of dipolar molecules in an optical lattice, taking into account all on-site and nearest-neighbor interactions, including occupation-dependent tunneling and pair tunneling terms. Using exact diagonalization and the multiscale entanglement renormalization ansatz, we show that these terms can destroy insulating phases and lead to novel quantum phases. These considerable changes of the phase diagram have to be taken into account in upcoming experiments with dipolar molecules.

Two-flavour mixture of a few fermions of different mass in a one-dimensional harmonic trap

A system of two species of fermions of different mass confined in a one-dimensional harmonic trap is studied with an exact diagonalization approach. It is shown independently on the number of particles that a mass difference between fermionic species induces a separation in the lighter flavor system. The mechanism of emerging of separated phases is explained phenomenologically and confirmed with the help of a direct inspection of the ground-state of the system. Finally, it is shown that the separation driven by a mass difference, in contrast to the separation induced by a difference of populations, is robust to the interactions with thermal environment.

Few interacting fermions in a one-dimensional harmonic trap

We study spin-1/2 fermions, interacting via a two-body contact potential, in a one-dimensional harmonic trap. Applying exact diagonalization, we investigate their behavior at finite interaction strength, and discuss the role of the ground-state degeneracy which occurs for sufficiently strong repulsive interaction. Even low temperature or a completely depolarizing channel may then dramatically influence the systems behavior. We calculate level occupation numbers as signatures of thermalization, and we discuss the mechanisms to break the degeneracy.

Pairing in a system of a few attractive fermions in a harmonic trap

We study a strongly attractive system of a few spin-(1/2) fermions confined in a one-dimensional harmonic trap, interacting via two-body contact potential. Performing exact diagonalization of the Hamiltonian we analyze the ground state and the thermal state of the system in terms of one- and two-particle reduced density matrices. We show how for strong attraction the correlated pairs emerge in the system. We find that the fraction of correlated pairs depends on temperature and we show that this dependence has universal properties analogous to the gap function known from the theory of superconductivity. In contrast to the standard approach based on the variational ansatz and/or perturbation theory, our predictions are exact and are valid also in a strong-attraction limit. Our findings contribute to the understanding of strongly correlated few-body systems and can be verified in current experiments on ultra-cold atoms.

Quantum electrodynamics of qubits

A systematic description of a spin one-half system endowed with magnetic moment or any other two-level system qubit interacting with the quantized electromagnetic field is developed. This description exploits a close analogy between a two-level system and the Dirac electron that comes to light when the two-level system is described within the formalism of second quantization in terms of fermionic creation and annihilation operators. The analogy enables one to introduce all the powerful tools of relativistic QED albeit in a greatly simplified form. The Feynman diagrams and the propagators turn out to be very useful. In particular, the QED concept of the vacuum polarization finds its close counterpart in the photon scattering off a two level system leading via the linear response theory to the general formulas for the atomic polarizability and the dynamic single spin susceptibility. To illustrate the usefulness of these methods, we calculate the polarizability and susceptibility up to the fourth order of perturbation theory. These ab initio calculations resolve some ambiguities concerning the sign prescription and the optical damping that arise in the phenomenological treatment. We also show that the methods used to study two-level systems qubits can be extended to many-level systems qudits. As an example, we describe the interaction with the quantized electromagnetic field of an atom with four relevant states: one S state and three degenerate P states. I. INTRODUCTION Two-level quantum systems, called qubits by Schumacher 1, play a fundamental role in quantum information theory. In this context they are usually treated as mathematical objects living in a two-dimensional Hilbert space. In reality, qubits always exist as material objects and we should not forget that they are endowed with concrete physical properties. In this paper we shall deal with two-level systems that interact directly with the electromagnetic field, such as spin one-half particles endowed with magnetic moment or twolevel atoms. Thus our results do not apply to qubits encoded in the polarization states of photons. We shall restrict ourselves in this paper to isolated qubits interacting only with the quantized electromagnetic field. Therefore the calculated decay rates will include only the spontaneous emission. A two-level system is the simplest model of a quantum system and yet in the presence of a coupling to the quantized electromagnetic field an exact solution has not been obtained. Even in the simplest case, when the electromagnetic field is restricted to just one mode, the model has been exactly solved only in the rotating-wave approximation by Jaynes and Cummings 2. Among the approximate solutions, perturbation theory is still the most universal and effective tool, especially in the world of electromagnetic phenomena. In the present paper we develop a systematic and complete theory based on an observation that a two-level system can be treated as a relativistic trapped electron. The translational degrees of freedom of such an electron are practically frozen. The only “degree of freedom” that remains is the electron’s ability to undergo transitions between two discrete energy states. In order to fully unfold the connection between the QED and the theory of two-level systems, we shall perform the second quantization of the standard theory of qubits. The description of two-level systems in terms of creation and annihilation operators has been introduced before cf., for example, 3 but no one has exploited the full potential of this formulation. The crucial element in our formulation is the systematic use of Feynman diagrams. To expose a close analogy with the relativistic theory, including the form of the propagators, we shall choose the energy scale in such a way that the energy levels of the two-level system have opposite signs. In this way, we arrive at a picture of a two-level system that coincides with the Dirac-sea view of quantum electrodynamics. The ground state of the two-level system corresponds to the occupation of the negative energy state, while the excited state corresponds to the occupation of the positive energy state accompanied by a hole in the negative energy sea. The transition between these two states due to the interaction with a photon can be represented by the two elementary Feynman diagrams shown in Fig. 1.

Exact diagonalization of the one-dimensional Bose-Hubbard model with local three-body interactions

In this Brief Report the extended Bose-Hubbard model with local two- and three-body interactions is studied by the exact diagonalization approach. The shapes of the first two insulating lobes are discussed and the values of the critical tunneling for which the insulating phase loses stability for repulsive and attractive three-body interactions are predicted.

Ground-state entanglement of spin-1 bosons undergoing superexchange interactions in optical superlattices

We will discuss a model with ultracold atoms confined in optical superlattices. In particular, we will study the ground-state properties of two spin-1 bosons trapped in a double-well potential. Depending on the external magnetic field and biquadratic interactions, different phases of magnetic order are realized. Applying von Neumann entropy and the number of relevant orbitals, we will quantify the bipartite entanglement between particles. Changing the values of the parameters determining the superlattices, we can switch the system between differently entangled states.

Creation on Demand of Higher Orbital States in a Vibrating Optical Lattice

It is shown that the extended Hubbard Hamiltonian describing atoms confined in an optical lattice always contains commonly neglected terms which can significantly change the dynamical properties of the system. Particularly for bosonic systems, they can be exploited for creating orbital states on demand via the parametric resonance phenomenon. This indicates an additional application for optical lattices, namely, the study and emulation of interactions between particles and lattice vibrations.

Few strongly interacting ultracold fermions in one-dimensional traps of different shapes

The ground-state properties of a few spin-1/2 fermions with different masses and interacting via short-range contact forces are studied within an exact diagonalization approach. It is shown that, depending on the shape of the external confinement, different scenarios of the spatial separation between components manifested by specific shapes of the density profiles can be obtained in the strong interaction limit. We find that the ground-state of the system undergoes a specific transition between orderings when the confinement is changed adiabatically from a uniform box to a harmonic oscillator shape. We study the properties of this transition in the framework of the finite-size scaling method adopted to few-body systems.