An optical lattice with cold trapped atoms represents a quantum system of fundamental importance as it enables the study of quantum many-body system in a controllable way. It is thus necessary to develop theoretical and experimental tools to explore quantum correlation in such systems to advance our understanding of many-body physics. While previous works have identified some profound aspects of quantum entanglement using e.g. entanglement entropy, there exists a critical demand to have an experimentally accessible tool to investigate many-body quantum entanglement in a broad context. We present an entanglement criterion characterizing collective entanglement in Boson lattice systems and enabling experimental observation readily. On applying our approach to the extended Bose-Hubbard model, we show that our criterion witnesses phase transitions such as Mott insulator--superfluid and Mott insulator--charge density wave transitions. Remarkably, it also makes it possible to detect multipartite entanglement among boson lattice sites in a rigorous sense. Our criterion can be experimentally tested via Raman scattering or time-of-flight methods, which are within the reach of current technology.
An optical lattice with cold trapped atoms represents a quantum system of fundamental importance as it enables the study of quantum many-body system in a controllable way. It is thus necessary to develop theoretical and experimental tools to explore quantum correlation in such systems to advance our understanding of many-body physics. While previous works have identified some profound aspects of quantum entanglement using e.g. entanglement entropy, there exists a critical demand to have an experimentally accessible tool to investigate many-body quantum entanglement in a broad context. We present an entanglement criterion characterizing collective entanglement in Boson lattice systems and enabling experimental observation readily. On applying our approach to the extended Bose-Hubbard model, we show that our criterion witnesses phase transitions such as Mott insulator--superfluid and Mott insulator--charge density wave transitions. Remarkably, it also makes it possible to detect multipartite entanglement among boson lattice sites in a rigorous sense. Our criterion can be experimentally tested via Raman scattering or time-of-flight methods, which are within the reach of current technology.
Entanglement is a fundamental resource for quantum information processing [1-3], its applications including scalable quantum communication, secure quantum key distribution protocols for cryptography, and exponential speedup of quantum algorithms. It occurs naturally in manybody systems at low temperatures: This makes cold atoms good candidates to provide entangled states involving a large number of particles. For realizing and manipulating such states, an essential task is quantifying the actual amount of entanglement contained in the created state before using it for quantum information protocols. However, this task represents a major challenge, as it requires either full state tomography, scaling exponentially in the system size, or the assumption of unverified system characteristics such as its Hamiltonian or temperature. In this experiment [4], we obtain a quantification of multipartite entanglement from readily accessible measurements - that is, time-of-flight atomic density distribution - based on a recent proposal [5]. We directly observe and quantify the multipartite spatial entanglement between the sites of a periodic optical potential (optical lattice) that hosts massive bosonic particles. We characterize its behaviour when crossing the superfluid-Mott insulator transition and when varying temperature, demonstrating the robustness of our method. References [1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (Cambridge, 2000). [2]R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865-942 (2009). [3]O. Göhne and G. Tóth, Entanglement detection, Physics Reports 474, 1-75 (2009). [4]M. Cramer, A. Bernard, N. Fabbri, L. Fallani, C. Fort, S. Rosi, F. Caruso, M. Inguscio, and M. B. Plenio, Entanglement of bosons in optical lattices, E-print arXiv:1302.4897 [5]M. Cramer, M. B. Plenio, and H. Wunderlich, Measuring entanglement in condensed matter systems, Phys. Rev. Lett. 106, 020401 (2011).
Experiments with ultracold atoms in optical lattices provide outstanding opportunities to realize exotic quantum states due to a high degree of tunability and control. In this thesis, I present experiments that extend this control from global parameters to the level of individual particles. \n\nUsing a quantum gas microscope for 87Rb, we have developed a single-site addressing scheme based on digital amplitude holograms. The system self-corrects for aberrations in the imaging setup and creates arbitrary beam profiles. We are thus able to shape optical potentials on the scale of single lattice sites and control the dynamics of individual atoms. \n\nWe study the role of quantum statistics and interactions in the Bose-Hubbard model on the fundamental level of two particles. Bosonic quantum statistics are apparent in the Hong-Ou-Mandel interference of massive particles, which we observe in tailored double-well potentials. These underlying statistics, in combination with tunable repulsive interactions, dominate the dynamics in single- and two-particle quantum walks. We observe highly coherent position-space Bloch oscillations, bosonic bunching in Hanbury Brown-Twiss interference and the fermionization of strongly interacting bosons.\n\nMany-body states of indistinguishable quantum particles are characterized by large-scale spatial entanglement, which is difficult to detect in itinerant systems. Here, we extend the concept of Hong-Ou-Mandel interference from individual particles to many-body states to directly quantify entanglement entropy. We perform collective measurements on two copies of a quantum state and detect entanglement entropy through many-body interference. We measure the second order Rényi entropy in small Bose-Hubbard systems and detect the buildup of spatial entanglement across the superfluid-insulator transition.\n\nOur experiments open new opportunities for the single-particle-resolved preparation and characterization of many-body quantum states.
We analyze the developing of bipartite and multipartite entanglement through the Mott insulator-superfluid quantum phase transition. Starting from a Mott insulator state, where a filling factor ν = N/M = 1 per lattice site is considered, we derive an expression for a completely connected graph configuration of bosons and show how bipartite and multipartite entanglement evolve through the phase transition predicted in previous works. We show how, in the transition, bipartite entanglement is distributed through the system respecting monogamous relations.
For systems of interacting, ultracold spin-zero neutral bosonic atoms, harmonically trapped and subject to an optical lattice potential, we derive an Extended Bose Hubbard (EBH) model by developing a systematic expansion for the Hamiltonian of the system in powers of the lattice parameters and of a scale parameter, the lattice attenuation factor. We identify the dominant terms that need to be retained in realistic experimental conditions, up to nearest-neighbor interactions and nearest-neighbor hoppings conditioned by the on-site occupation numbers. In the mean field approximation, we determine the free energy of the system and study the phase diagram both at zero and at finite temperature. At variance with the standard on site Bose Hubbard model, the zero-temperature phase diagram of the EBH model possesses a dual structure in the Mott insulating regime. Namely, for specific ranges of the lattice parameters, a density wave phase characterizes the system at integer fillings, with domains of alternating mean occupation numbers that are the atomic counterparts of the domains of staggered magnetizations in an antiferromagnetic phase. We show as well that in the EBH model, a zero-temperature quantum phase transition to pair superfluidity is, in principle, possible, but completely suppressed at the lowest order in the lattice attenuation factor. Finally, we determine the possible occurrence of the different phases as a function of the experimentally controllable lattice parameters.
We analyze the ground-state entanglement entropy of the extended Bose-Hubbard model with infinite-range interactions. This model describes the low-energy dynamics of ultracold bosons tightly bound to an optical lattice and dispersively coupled to a cavity mode. The competition between on-site repulsion and global cavity-induced interactions leads to a rich phase diagram, which exhibits superfluid, supersolid, and insulating (Mott and checkerboard) phases. We use a slave-boson treatment of harmonic quantum fluctuations around the mean-field solution and calculate the entanglement entropy across the phase transitions. At commensurate filling, the insulator-superfluid transition is signaled by a singularity in the area-law scaling coefficient of the entanglement entropy, which is similar to the one reported for the standard Bose-Hubbard model. Remarkably, at the continuous Z_{2} superfluid-to-supersolid transition we find a critical logarithmic term, regardless of the filling. This behavior originates from the appearance of a roton mode in the excitation and entanglement spectrum, becoming gapless at the critical point, and it is characteristic of collective models.
In this dissertation, I present my work on ground state entanglement in Bose-Hubbard-type models. Bose-Hubbard-type models form a wide class of bosonic lattice models and describe systems that can be realized by cold atoms or molecules trapped in optical lattices. By studying ground state entanglement, one can understand quantum phases and phase transitions among them from a microscopic point of view. I will first introduce the two-component Bose-Hubbard model describing Bose-Bose mixtures, and discuss the role played by the interspecies entanglement in the Mott-insulator-to-superfluid phase transition of one component in the presence of a second component. I will show that interspecies entanglement provides a new perspective to understand quantum phases in mixtures. Then I will discuss long-range spatial entanglement in single-species Bose-Hubbard-type models. Specifically, I will build a generic framework for understanding under which conditions these models harbor certain types of topological order. My results provide guidance for future numerical studies which can pave the way for searching experimentally realizable bosonic lattice models harboring nontrivial topological order.