Quantum physics: Getting the measure of entanglement.

Ultracold atoms in optical lattices provide an extremely clean and controllable setting to explore quantum many-body phases of matter. Here we give a brief review of the strong-correlation physics that has been realized using such ultracold atoms in optical lattices ranging from the realization of Hubbard models to studies of quantum magnetism and the detection of single atoms with lattice site resolution. All this has opened up fundamentally new opportunities for the investigation of quantum many-body systems. Introduction Over the past years, ultracold atoms in optical lattices have emerged as versatile new systems to explore the physics of quantum many-body systems. On the one hand, they can be helpful in gaining a better understanding of known phases of matter and their dynamical behavior; on the other hand, they allow one to realize completely novel quantum systems that have not been studied before in nature [1, 2, 3]. Commonly, the approach of exploring quantum many-body systems in such a way is referred to as “quantum simulations.” Examples of some of the first strongly interacting many-body phases that have been realized both in lattices and in the continuum include the quantum phase transition from a superfluid to a Mott insulator [4, 5, 6], fermionic Mott insulators [7, 8], the achievement of a Tonks- Girardeau gas [9, 10], and the realization of the Bose-Einsten condensate (BEC)– Bardeen-Cooper-Schrieffer (BCS) crossover (see also Chapter 12) in Fermi gas mixture [11] using Feshbach resonances [12]. In almost all of these experiments, detection was limited to time-of-flight imaging or more refined derived techniques that mainly characterized the momentum distribution of the quantum gas [2]. For several years, researchers in the field have therefore aspired to employ in situ single-particle detection methods for the analysis of ultracold quantum gases. Only recently has it become possible to implement such imaging techniques, marking a milestone for the characterization and manipulation of ultracold quantum gases [13, 14, 15, 16, 17]. In our discussion, we will focus on one of the most successful of these techniques based on high-resolution fluorescence imaging. Despite being a rather new technique, such quantum gas microscopy has already proven to be an enabling technology for probing and manipulating quantum many-body systems. For the first time, controllable and strongly interacting many-body systems, as realized with ultracold atoms, could be observed on a local scale [17, 16].

Research focusing on the formation of ultracold atomic and molecular quantum gases is a continuously expanding field due to its envisioned applications such as quantum-controlled chemistry or quantum simulation. The aim of our theoretical work is to find ways to suppress inelastic or reactive processes between colliding particles in ultracold quantum gases. Using a laser field detuned to the blue of a relevant transition, we propose to couple the initial colliding particle state to a repulsive excited one, thus preventing the particles to come close to each other. This "blueshielding" technique is applied for ultracold quantum gases, close to quantum degeneracy. We investigate the optical shielding in ultracold <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">39</sup> K-Cs collision by using close-coupling channel method. As the (3) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> Σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> and (2) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> Π <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> molecular states are repulsive in long range[1], one verified effective optical shielding can be realized via driving the transition from the ground α <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> Σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> state to these excited states in the presence of linear-polarized light. We demonstrate that a strong field with a small blue detuning to the atomic transition is to the benefit of shielding since the Condon point is farther away from the inner region of ground triplet state. The spontaneous emission effect is taken into account by considering excited optical potentials with the modified formula of shielding rate [2]. We expect the shielding efficiency could be improved by using circular-polarized light [3].

This article reviews recent progresses in ultracold quantum gases, and it includes three subjects which are the Fermi gases across a Feshbach resonance, quantum gases in the optical lattices and the fast rotating quantum gases. In this article, we discuss many basic physics pictures and concepts in quantum gases, for examples, the resonant interaction, universality and condensation in the lowest Landau level; we introduce fundamental theoretical tools for studying these systems, such as mean-field theory for BEC-BCS crossover and for the boson Hubbard model; also, we emphasize the important unsolved problems in the forefront of this field, for instance, the temperature effect in optical lattices.

A topological charge pump is a one-dimensional, dynamical analogue of a quantum Hall system that is particularly well suited for studying topology, both theoretically and experimentally, since no synthetic gauge fields are required and efficient numerical algorithms exist to treat one-dimensional interacting systems in the form of matrix-product state methods. Instead of a robustly quantized Hall conductivity, a topological charge pump pumps a quantized amount of charge in each cycle of a parameter modulation that is given by the same topological invariant as in the quantum Hall effect. The goal of this thesis is to understand the stability of quantized transport in various contexts that are relevant in ultracold quantum gases, such as disordered systems, quantum systems where energy can be exchanged with a bath and in the presence of genuine many-body interactions. The results are presented as part of four scientific papers. In the first paper, we characterize the breakdown of quantized pumping as a function of disorder strength by employing various instantaneous and time-dependent measures. We argue that in disorder averages, the full energy gap distributions should be considered. In the second paper, we study the breakdown of quantized pumping due to electron-phonon coupling and find a resonant breakdown when the pump frequency equals the phonon frequency. The third paper introduces the possibility of a splitting of critical points in a Rice-Mele Hubbard model, which allows for interaction-induced pumping. The final paper presents experimental data of an ultracold-atom setup studying this interaction-induced pumping, as well as numerical calculations detailing the breakdown due to the crossing of a spin-gapless line.

Entanglement is the fundamental difference between classical and quantum\nsystems and has become one of the guiding principles in the exploration of\nhigh- and low-energy physics. The calculation of entanglement entropies in\ninteracting quantum field theories, however, remains challenging. Here, we\npresent the first method for the explicit computation of reduced density\nmatrices of interacting quantum field theories using truncated Hamiltonian\nmethods. The method is based on constructing an isomorphism between the Hilbert\nspace of the full system and the tensor product of Hilbert spaces of\nsub-intervals. This naturally enables the computation of the von Neumann and\narbitrary R\\'enyi entanglement entropies as well as mutual information,\nlogarithmic negativity and other measures of entanglement. Our method is\napplicable to equilibrium states and non-equilibrium evolution in real time. It\nis model independent and can be applied to any Hamiltonian truncation method\nthat uses a free basis expansion. We benchmark the method on the free\nKlein-Gordon theory finding excellent agreement with the analytic results. We\nfurther demonstrate its potential on the interacting sine-Gordon model,\nstudying the scaling of von Neumann entropy in ground states and real time\ndynamics following quenches of the model.\n

Linear response theory of entanglement entropy

After deriving the isothermal Hellmann-Feynman theorem (IHFT) that is suitable for mixed states in a canonical ensemble, we use this theorem to obtain the isothermal magnetic-field sweep theorems for the free, average, and trapping energies and for the entropy, specific heat, pressure, and atomic compressibility of strongly correlated ultracold quantum gases. In particular, we apply the sweep theorems to two-component Fermi gases in the weakly interacting Bardeen-Cooper-Schrieffer and Bose-Einstein condensate limits, showing that the temperature dependence of the contact parameter can be determined by varying either the entropy or specific heat with respect to the scattering length. We also use the IHFT to obtain the virial theorem in a canonical ensemble and discuss its implications for quantum gases.

In quantum information theory, the entropy of entanglement is a standard measure of bipartite entanglement between two partitions of a composite system. For a particular class of continuous variable quantum states, the Gaussian states, the entropy of entanglement can be expressed elegantly in terms of symplectic eigenvalues, elements that characterise a Gaussian state and depend on the correlations of the canonical variables. We give a rigorous step-by-step derivation of this result and provide physical insights, together with an example that can be useful in practice for calculations.

In this paper, we consider two extensions of the Gacs–Korner common information to three variables, the conditional common information (cCI) and the coarse-grained conditional common information (ccCI). Both quantities are shown to be useful technical tools in the study of classical and quantum resource transformations. In particular, the ccCI is shown to have an operational interpretation as the optimal rate of secret key extraction from an eavesdropped classical source $p_{XYZ}$ when Alice ( $X$ ) and Bob ( $Y$ ) are unable to communicate but share common randomness with the eavesdropper Eve ( $Z$ ). Moving to the quantum setting, we consider two different ways of generating a tripartite quantum state from classical correlations $p_{XYZ}$ : 1) coherent encodings $\sum _{xyz}\sqrt {p_{xyz}} |xyz\rangle $ and 2) incoherent encodings $\sum _{xyz}p_{xyz} |xyz\rangle \langle xyz|$ . We study how well can Alice and Bob extract secret key from these quantum sources using quantum operations compared with the extraction of key from the underlying classical sources $p_{XYZ}$ using classical operations. While the power of quantum mechanics increases Alice and Bob’s ability to generate shared randomness, it also equips Eve with a greater arsenal of eavesdropping attacks. Therefore, it is not obvious who gains the greatest advantage for distilling secret key when replacing a classical source with a quantum one. We first demonstrate that the classical key rate of $p_{XYZ}$ is equivalent to the quantum key rate for an incoherent quantum encoding of the distribution. For coherent encodings, we next show that the classical and quantum rates are generally incomparable, and in fact, their difference can be arbitrarily large in either direction. Finally, we introduce a “zoo” of entangled tripartite states all characterized by the conditional common information of their encoded probability distributions. Remarkably, for these states almost all entanglement measures, such as Alice and Bob’s entanglement cost, squashed entanglement, and relative entropy of entanglement, can be sharply bounded or even exactly expressed in terms of the conditional common information. In the latter case, we thus present a rare instance in which the various entropic entanglement measures of a quantum state can be explicitly calculated.

Nonlocal realism tests and quantum state tomography in Sagnac-based type-II polarization-entanglement SPDC-source

A considerable body of experimental and theoretical work claims the existence of negative absolute temperatures in spin systems and ultra-cold quantum gases. Here, we clarify that such findings can be attributed to the use of a popular yet inconsistent entropy definition, which violates fundamental thermodynamic relations and fails to produce sensible results for simple analytically tractable classical and quantum systems. Within a mathematically consistent thermodynamic formalism, based on an entropy concept originally derived by Gibbs, absolute temperature remains positive even for systems with bounded spectrum. We address spurious arguments against the Gibbs formalism and comment briefly on heat engines with efficiencies greater than one.

Scattering resonances are fundamental to many areas of physics, occurring across a wide range of energy scales. In ultracold quantum gases, magnetic Feshbach resonances have transformed quantum many-body research by enabling precise interaction control between atoms. Here, we demonstrate unprecedented control to engineer Feshbach resonances at tunable positions via Floquet driving in a lithium-6 (^{6}Li) atom gas, achieved through strong magnetic field modulation at megacycles per second frequencies. This periodic modulation creates scattering resonances whenever dressed molecular levels cross the atomic threshold. By adding a second modulation at twice the base frequency, we tune the asymmetry of resonance loss profiles and suppress two-body losses from Floquet heating. This technique enhances control over atomic interactions, expanding possibilities for quantum simulations of complex systems and studies of exotic quantum phases.

We calculated the von Neumann entanglement entropy and the Schmidt number of one dimentional (1D) cluster states and showed that these are useful measures to estimate entanglement caused by delocalization of clusters. We analyze system size dependence of these entanglement measures in the linear-chain $n\alpha$ states given by Tohsaki-Horiuchi-Schuck-R\"opke wave functions for 1D cluster gas states. We show that the Schmidt number is an almost equivalent measures to the von Neumann entanglement entropy when the delocalization of clusters occurs in the entire system but it shows different behaviors in a partially delocalized state containing localized clusters and delocalized ones. It means that the R\'enyi-2 entanglement entropy, which relates to the Schmidt number, is found to be almost equivalent to the von Neumann entanglement entropy for the full delocalized cluster system but it is less sensitive to the partially delocalized cluster system than the von Neumann entanglement entropy. We also propose a new entanglement measure which has a generalized form of the Schmidt number. Sensitivity of these measures of entanglement to the delocalization of clusters in low-density regions was discussed.

With rapid progress across platforms for quantum systems, the problem of many-body quantum state reconstruction for noisy quantum states becomes an important challenge. There has been a growing interest in approaching the problem of quantum state reconstruction using generative neural network models. Here we propose the ‘attention-based quantum tomography’ (AQT), a quantum state reconstruction using an attention mechanism-based generative network that learns the mixed state density matrix of a noisy quantum state. AQT is based on the model proposed in ‘Attention is all you need’ by Vaswani et al (2017 NIPS) that is designed to learn long-range correlations in natural language sentences and thereby outperform previous natural language processing (NLP) models. We demonstrate not only that AQT outperforms earlier neural-network-based quantum state reconstruction on identical tasks but that AQT can accurately reconstruct the density matrix associated with a noisy quantum state experimentally realized in an IBMQ quantum computer. We speculate the success of the AQT stems from its ability to model quantum entanglement across the entire quantum system much as the attention model for NLP captures the correlations among words in a sentence.

We discuss several issues important for experimentally observing Efimov physics in ultracold quantum gases. By numerically solving the three-boson Schrödinger equation over a broad range of scattering lengths and energies, and by including model potentials with multiple bound states, we address the complications of relating experimental observations to available analytic expressions. These more realistic potentials introduce features that can mask the predicted Efimov physics at small scattering lengths. They also allow us to verify that positive and negative scattering lengths are universally connected only across a pole, not across a zero. Additionally, we show that the spacing between Efimov features for the relatively small scattering lengths accessible experimentally fails to precisely follow the geometric progression expected for Efimov physics. Finally, we emphasize the importance of the short-range three-body physics in determining the position of Efimov features and show that theoretically reproducing two-body physics is not generally sufficient to predict three-body properties quantitatively.