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Quantum optics and frontiers of physics: The third quantum revolution Alessio Celi,1 Anna Sanpera,2, 3 Veronica Ahufinger,3 and Maciej Lewenstein1, 2, ∗

arXiv:1601.04616v2 [cond-mat.quant-gas] 10 Jul 2016

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ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Av. C.F. Gauss 3, 08860 Castelldefels, Spain 2 ICREA – Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain 3 Departament de Física, Universitat Autònoma Barcelona, 08193 Bellaterra, Spain The year 2015 was the International Year of Light. It marked, however, also the 20th anniversary of the first observation of Bose-Einstein condensation in atomic vapors by Eric Cornell, Carl Wieman and Wolfgang Ketterle. This discovery can be considered as one of the greatest achievements of quantum optics that has triggered an avalanche of further seminal discoveries and achievements. For this reason we devote this essay for the focus issue on “Quantum Optics in the International Year of Light” to the recent revolutionary developments in quantum optics at the frontiers of all physics: atomic physics, molecular physics, condensed matter physics, high energy physics and quantum information science. We follow here the lines of the introduction to our book “Ultracold atoms in optical lattices: Simulating quantum many-body systems” [1]. The book, however, was published in 2012, and many things has happened since then – the present essay is therefore upgraded to include the latest developments. PACS numbers: 42.50.-p, 03.67.-a

The achievement of Bose-Einstein condensation (BEC) in dilute gases in 1995 [2–4] marks the beginning of a new era. For the so-called AMO community —comprising atomic, molecular, optics and quantum optics— it became soon evident that condensed matter physics, i.e., degenerate quantum many body systems could be at reach. The condensed matter community remained at that stage more skeptical. They argued that, at the very end, what was achieved experimentally was a regime of weakly interacting Bose gases; the domain thoroughly investigated by the condensed matter theorists in the 50’s and 60’s [5, 6]. For solid state/condensed matter experts the fact that the AMO experiments dealt with confined systems of finite size and typically inhomogeneous densities was a technical issue rather than a question of fundamental importance. Nonetheless, the Nobel foundation awarded its yearly prize in 2001 to E. A. Cornell, C. E. Wieman and W. Ketterle “for the achievement of BoseEinstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates” [7, 8]. Today, from the perspective of some years, we see that due to the efforts of the whole community these fundamental studies have enriched amazingly the standard “condensed matter” understanding of static and dynamical properties of weakly interacting Bose gases [9]. From the very beginning the AMO community continued their efforts to push the BEC physics towards new regimes and new challenges. The progress in these directions was indeed spectacular, and at the beginning of the third millennium became clear to both, AMO and condensed matter communities, that indeed we are entering a truly new quantum era with unprecedented possibilities of control of many body systems. In particular, it became clear that the regime of strongly correlated

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systems may be reached with ultracold atoms and/or molecules. Few years after the first observation of BEC, in 1999 atomic degenerate Fermi gases were achieved [10–12] paving the way toward the observations of Fermi superfluidity. This is described, in the weak interaction limit by the Bardeen–Cooper–Schrieffer theory (BCS) [6], and by the so called BEC-BSC crossover in the limit of strong correlations. For recent reviews reporting the large activity in this field see [13] and [14]. Even earlier, following the seminal proposal by Jaksch et al. [15], Greiner et al. [16] observed the signatures of the quantum phase transition from the superfluid to the, so called Mott insulator state for bosons confined in an optical lattice. Nowadays, ultracold atomic and molecular systems are at the frontiers of modern quantum physics, and are considered to provide the most controllable systems to study many body physics. It is believed that these systems will also find highly nontrivial applications in quantum information, and quantum metrology and will serve as powerful quantum simulators. At the theory level, the broadness of the different fields touched by ultracold atoms has lead to a “grand unification” where AMO, condensed matter, nuclear physics, and even high energy physics theorists are joining efforts —for reviews see [17] and [14]—. After the first quantum revolution at the beginning of the XX century, the second one associated with the name of John Bell and the experimental quest for non-locality of quantum mechanics together with the experimental control over single, or few particle systems —see in particular Alain Aspect’s introduction to [18]—, we are witnessing the third one: the quantum revolution associated to the control over macroscopic quantum systems and the rise of quantum technologies. Indeed this quantum revolution is not limited to the ultracold atoms but includes other platforms like NV-centers [19] and superconducting qubits [20] where quantum optics also plays a central role.

2 I.

COLD ATOMS FROM A HISTORICAL QUANTUM OPTICAL PERSPECTIVE

The third quantum revolution does not arrive completely unexpected: the developments of atomic physics and quantum optics in the last 30 years have inevitably led to it. In the seventies and eighties of the XX century, atomic physics was a very well established and respectful area of physics. It was, however, by no means a “hot” area. On the theory side, even though one had to deal with complex problems of many electron systems, most of the methods and techniques were already developed. The major problems that were considered concerned mainly questions related on optimisation of these methods. These questions were reflecting an evolutionary progress, rather than a revolutionary search for totally new phenomena. On the experimental side, quantum optics at that time was entering its Golden Age. The development of laser physics and nonlinear optics led in 1981 to the Nobel prize for A.L. Schawlow and N. Bloembergen “for their contribution to the development of laser spectroscopy”. On the other hand, studies of quantum systems at the single particle level culminated in 1989 with the Nobel prize for H.G. Dehmelt and W. Paul “for the development of the ion trap technique”, shared with N.F. Ramsey ”for the invention of the separated oscillatory fields method and its use in the hydrogen maser and other atomic clocks”. Theoretical quantum optics was born in the 60’ties with the works on quantum coherence theory developed the 2005 Noble prize winner, R.J. Glauber [21, 22], and with the development of the laser theory in the late 60’ties by H. Haken and by M.O. Scully and W.E. Lamb (Nobel laureate of 1955). In the 70’ties and 80’ties, however, theoretical quantum optics was not considered to be a separate, established area of theoretical physics. One of the reasons of this state of the art, was that indeed quantum optics at that time was primarily dealing with single particle problems. Most of the many body problems of quantum optics, such as laser theory, or more generally optical instabilities [23], could have been solved either using linear models, or employing relatively simple versions of the mean field approach. Perhaps the most sophisticated theoretical contributions concerned understanding of quantum fluctuations and quantum noise [23, 24]. This situation has drastically changed due to the unprecedented level of quantum engineering, i.e. preparation, manipulation, control and detection of quantum systems developed and achieved by atomic physics and quantum optics. There are several seminal discoveries that have triggered these changes: • The cooling and trapping of atoms, ions and molecules have reached regimes of very low temperatures (today down to nano-Kelvin!) and precision that were considered unattainable just two decades ago. These developments were recognized by the Nobel Foundation in 1997, who

awarded the Prize to S. Chu [25], C. CohenTannoudji [26] and W.D. Phillips [27] “for the development of methods to cool and trap atoms with laser light”. Laser cooling and mechanical manipulations of particles with light [28] was essential for development of completely new areas of atomic physics and quantum optics, such as atom optics [29], and for reaching new territories of precision metrology and quantum engineering. • Laser cooling combined with evaporation cooling allowed, in 1995, the experimental observation Bose-Einstein condensation (BEC) [2–4], a phenomenon predicted by S. Bose and A. Einstein more than 70 years earlier. As we said before, these experiments marked evidently the birth of a new era. E.A. Cornell and C.E. Wieman [7] and W. Ketterle [8] received the Nobel Prize in 2001, “for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates”. This was truly a breakthrough moment, in which “atomic physics and quantum optics has met condensed matter physics” [8]. Condensed matter community at that time remained, however, still reserved. After all, BEC was observed in weakly interacting dilute gases, where it is very well described by the mean field Bogoliubov-de Gennes theory [9], developed for homogeneous systems in the 50’ties. • The study of quantum correlations and entanglement. The seminal theoretical works of the late A. Peres [30, 31], the proposals of quantum cryptography by C. H. Bennett and G. Brassard [32], and A. K. Ekert [33], the quantum communication proposals by C. H. Bennett and S. J. Wiesner [34] and C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters [35], the discovery of the quantum factorizing algorithm by P. Shor [36], and the quantum computer proposal by J. I. Cirac and P. Zoller [37] gave birth to the field of quantum information [38, 39] (for the latest version of the quantum information road map see [40]). These studies, together with the rapid development of the theory have led to an enormous progress in our understanding of quantum correlations and entanglement and, in particular, how to prepare and use entangled states as a resource. The impulses from quantum information enter nowadays constantly into the physics of cold atoms, molecules and ions, and stimulate new approaches. It is very probable that the first quantum computers will be, as suggested already by Feynman [41, 42], computers of special purpose – quantum simulators (QS) [43], that will efficiently simulate quantum many body systems that otherwise cannot be simulated using “classical” computers [39]. • Optical lattices and Feshbach resonances.

The

3 physics of ultracold atoms entered the area of strongly correlated systems with the seminal proposal of Jaksch et al. [15] on how to achieve the transition superfluid-Mott insulator transition in cold atoms using optical lattices. The proposal was based on the bosonic Mott insulator transition proposed in condensed matter [44], but the authors of this paper were in fact motivated by the possibility of realizing quantum computing with cold atoms in a lattice. Transition to the Mott insulator state was supposed to be an efficient way of preparing a quantum register with a fixed number of atoms per lattice site. Entanglement between atoms could be achieved via controlled collisions [45]. The experimental observation of the superfluid-Mott insulator transition by the Bloch–H´’ansch group [16] was without any doubts a benchmark in the studies of strongly correlated systems with ultracold atoms [46]. Several other groups have observed bosonic superfluid-Mott insulator transitions in pure Bose systems [47], in disordered Bose systems [48], in Bose-Fermi [49, 50], and Bose-Bose mixtures [51]. The possibility to change the collision properties of ultracold atoms by means of tuning the Feschbach resonances of the atomic species has been another tool of inestimable value. Such a tool has lead to the fermionic Mott insulator [52, 53]. Also, Mott insulator state of molecules have been created [54], as well as bound repulsive pairs of atoms (i.e. pairs of atoms at a site that cannot release their repulsive energy due to the band structure of the spectrum in the lattice) have been observed [55].

• Cold trapped ions were initially investigated to realize the Cirac-Zoller quantum gate [37], and to attempt to build an scalable quantum computer using a bottom-up approach [56, 57]. Recent progress in this research line can be found in [58]. A new stream of ideas using cold ions for quantum simulators have recently appeared. First proposals have shown that ion-ion interactions mediated by phonons can be manipulated to implement various spin models [59–61], where the spin states correspond to internal states of the ion. In tight linear traps, one could realize the spin chains with interactions decaying as (distance)−3 , i.e. as in the case of dipole-dipole interactions. Such spin chains may serve as quantum neural network models and may be used for adiabatic quantum information processing [62, 63]. More interestingly, ions can be employed as quantum simulators, both in 1D and in 2D, where the ions form self-assembled Coulomb crystals [64]. First steps towards the experimental realization of these ideas has been reported [65]. These results have been recently extended in [66–71] Alternatively to spins, one

could look at phonons in ion self-assembled crystals; these are also predicted to exhibit interesting collective behaviour from Bose condensation to strongly correlated states [72, 73]. Trapped ions may be used also to simulate mesoscopic spinboson models [74]. Combined with optical traps or ion microtrap array methods can be used to simulate a whole variety of spin models with tunable interactions in a wide range of spatial dimensions and geometries [75]. Finally, they are promising candidates for the realization of many-body interactions [76]. All these theoretical proposals and the spectacular progress in the experimental trapped ion community pushes trapped ions physics to be soon used as widely as cold atoms to mimic condensed matter physics and beyond, as for instance in [77].

II.

QUANTUM SIMULATORS

Before we proceed we would like to make a short detour to explain in a little more detail the concept of quantum simulators. XX century was the age of information and computers. But, even supercomputers have their limitations – as we well know for instance their ability to predict weather, a paradigmatic classical chaotic phenomenon, is very restricted. For this reason already in classical computer science the concept of special purpose computers was developed. These “classical simulators” are not like universal classical computers – they can only simulate or calculate certain restricted class of models describing Nature. The best simulations of classical disordered systems, such as spin glasses, are nowadays obtained with such computers of special purpose – “classical simulators”. The situation is even more dramatic in quantum physics and chemistry. We know very well that an universal quantum computer will revolutionize our technology in the future. Unfortunately, this future does not seem to be very close at hand, and definitely concerns tens of years. However, quantum computers with a special purpose, i.e. quantum simulators, exist since already 5 years (cf. [78]). Various quantum phenomena such as high-T c superconductivity or quark confinement are still awaiting universally accepted explanations because of the computational complexity of solving the simplified theoretical models designed to capture the relevant physics. Richard Feynman suggested in 1982 [41, 42], solving such models by ”quantum simulation”. He pointed out that it might be possible sometimes to find a simpler and experimentally more accessible system to mimic the quantum system of interest. Feynman’s idea was motivated by the complexity of classical simulations of quantum systems. Suppose we want to study a system of N spins 1/2; then the dimension of the Hilbert space is 2N , and the number of coefficients we need to describe

4 the wave function of the system may be in principle just as large. As N grows, this number quickly becomes larger than the number of atoms in the universe, so classical simulations are evidently impossible. Of course, in practice this number may be very much reduced by using clever representations of the wave functions, but in general there are many quantum systems, which are very hard to simulate classically. The modern concept of quantum simulators is not exactly the same as that of Feynman. Since condensed matter systems are typically very complex, theorists construct simplified models. In this simplification, symmetries of the original problem are kept intact, and the concept of universality is used: different Hamiltonians with similar symmetry properties belong to the same universality classes, i.e. they exhibit the same phase transitions and have the same critical exponents. Unfortunately, even these simplified models are often difficult to understand. A paradigmatic example of such a situation concerns high-T c superconductivity of cuprates, where it is believed that the basic physics is captured by an array of weakly coupled 2D Hubbard models for electrons, i.e. spin-1/2 fermions. Even this simple model reduced to one 2D plane does not allow for accurate treatment, and there is much controversy concerning, for instance, the phase diagram or the character of the transitions. The role of quantum simulators, as proposed in many quantum information projects, will be to simulate simple models, such as 2D Hubbard models, and obtain a better understanding of them, rather than try to simulate the full complexity of the condensed matter. A “working” definition of a quantum simulator could be as follows [1, 79]: • A quantum simulator is an experimental system that mimics a simple model, or a family of simple models, of condensed matter (or high-energy physics, or quantum chemistry...). • The simulated models have to be of some relevance for applications and/or our understanding of the challenges of the above-mentioned areas of physics, and of the new challenges in Physics, Chemistry and Biology that are still to be identified. • The simulated models should be computationally intractable for classical computers. Note that this statement may have two meanings: i) an efficient (scalable to large system size) algorithm to simulate the model might not exist, or might not be known; ii) the efficient scalable algorithm may be known, but the size of the simulated model is too large to be simulated under reasonable time and memory restrictions. The latter situation, in fact, occurs with classical simulations of the Bose or Fermi Hubbard models as compared to their experimental quantum simulators. There might also be exceptions to the general rule. For instance,

it is desirable to realize quantum simulators to simulate and to observe novel, hitherto only theoretically predicted phenomena, even though it might be possible to simulate these phenomena efficiently with present-day computers. Simulating and observing is more than just simulating. • A quantum simulator should allow for broad control of the parameters of the simulated model, and for control of the preparation, manipulation and detection of the states of the system. In particular, it is important to be able to set the parameters in such a way that the model becomes tractable using classical simulations. This provides the possibility of validating the quantum simulator. Fueled by the prospect of solving a broad range of long-standing problems in strongly-correlated systems, the tools to design, build, and implement QSs [1, 41, 79] have rapidly developed and are now reaching very sophisticated levels [78]. There exist many proposals and already many realizations of quantum simulators employing ultracold atoms and molecules in traps and in optical lattices (cf. [80–91], for a review see also [14]), probing quantum dynamics (cf. [92, 93]), employing ultracold trapped ions (cf. [65, 67, 94–98], for recent reviews see [99, 100]), atoms in single or in arrays of cavities (cf. [101]), ultracold atoms/ions in arrays of traps, ultracold atoms near nano-structures and multidimensional plasmonic traps [102–105], photons [106–110], arrays of quantum dots, circuit quantum electrodynamics (QED) and polaritons (cf. [111–115], for a recent review see [116]), artificial lattices in solid state [117], nuclear magnetic resonance (NMR) systems [118–121] and superconducting qubits. In addition to our book [1] there exist excellent recent reviews, in particular in the focus issue of Nature Physics Insight [122–127]. At the current pace, it is expected that we will soon reach the ability to finely control manybody systems whose description is outside the reach of a classical computer. For example, modeling interesting physics associated with a quantum system involving 50-100 spin-1/2 particles – whose general description requires 250 − 2100 ≈ 1015 − 1030 amplitudes – is out of the reach of current classical supercomputers, but perhaps within the grasp of a quantum simulators. Obviously, the real-world implementations of a quantum simulation will always face experimental imperfections, such as noise due to finite precision instruments and interactions with the environment. The quantum simulator – as envisioned by Feynman – is fundamentally an analog device, in the sense that all operations are carried out continuously. However, errors in an analog device (also continuous, like temperature in the initial state, or the signal-to-noise ratio of measurement) can propagate and multiply uncontrollably [128]. Indeed, Landauer, a father of the studies of the physics of information, questioned whether quantum coherence was truly a powerful resource for computation because

5 it required a continuum of possible superposition states that were “analog” in nature [129]. In contrast to digital quantum simulators, where the error correction schemes might be in principle applied, the question of validation, calibration and reliability of analog quantum simulators is very subtle and very ”hot” [130]. III.

COLD ATOMS AND QUANTUM OPTICS AT THE FRONTIERS OF PHYSICS

Quantum optics and physics of cold atoms touches nowadays, among others, common frontiers of contemporary physics with condensed matter physics, quantum many body physics, nuclear physics, quantum field theory, high energy physics, and even astrophysics. In particular, many important challenges of the latter disciplines can be addressed within cold atoms and quantum optical wizardry: • 1D systems. The role of quantum fluctuations and correlations is particularly important in 1D. The theory of 1D systems is very well developed due to the existence of exact methods such as Bethe Ansatz and quantum inverse scattering theory (cf. [131]), powerful approximate approaches, such as bosonization, or conformal field theory —cf. [132]—, and efficient computational methods, such as density matrix Renormalization group technique (DMRG) —cf. [133]—. There are, however, many open experimental challenges that have not been so far directly and clearly realized in condensed matter, and can be addressed with cold atoms —for a review see [134, 135]—. Examples include atomic Fermi, or Bose analogues of spin-charge separation, or more generally observations of microscopic properties of Luttinger liquids [136–138]. Experimental observations of the 1D gas in the deep Tonks-Girardeau regime by Paredes et al. [139] —see also [140], [141], [47] and [142]— were the first steps in this direction. The recent achievement of Tonks regime in 2008 employing dissipative processes (two body losses) is perhaps the first experimental example of how to control a system by making use of its coupling to the environment [143–146]. See also the recent progress towards deep Tonks-Girardeau regime in [147]. • Spin-boson model. A two-level system coupled to a bosonic reservoir is a paradigmatic model both, in quantum dissipation theory in quantum optics and as quantum phase transition in condensed matter where it is termed as the spin-boson model —for a review see [148]—. It has also been proposed [149] that an atomic quantum dot, i.e., a single atom in a tight optical trap coupled to a superfluid reservoir via laser transitions, may realize this model. In particular, atomic quantum dots embedded in

a 1D Luttinger liquid of cold bosonic atoms accomplishes a spin-boson model with Ohmic coupling, which exhibits a dissipative phase transition and allows to directly measure atomic Luttinger parameters. Also, it has been shown that in the low-energy limit the system of a driven quantum spin coupled to a bath of ultracold fermions can be mapped onto the spin-boson model with an Ohmic bath [150] • 2D systems. According to the celebrated MerminWagner-Hohenberg theorem, 2D systems with continuous symmetry do not exhibit long range order at temperatures T > 0. 2D systems may, however, undergo Kosterlitz-Thouless-Berezinskii transition (KTB) to a state in which the correlations decay is algebraic, rather than exponential. Although KTB transition has been observed in liquid Helium [151, 152], its microscopic nature (binding of vortex pairs) has never been seen. Recent experiments [153–159] make an important step in this direction. • Hubbard and spin models. Many very important examples of strongly correlated states in condensed matter physics are realized in various types of Hubbard models [131, 160]. While Hubbard models in condensed matter are “reasonable caricatures” of real systems, ultracold atomic gases in optical lattices allow to achieve practically perfect realizations of a whole variety of Hubbard models [161]. Similarly, in certain limits Hubbard models reduce to various spin models; again cold atoms and ions allow for practically perfect realizations of such spin models —see for instance [162–165] and [66, 68, 70, 71, 87]—. Specifically, spin-spin interactions between neighboring atoms can be implemented by bringing the atoms together on a single site and carrying out controlled collisions [45, 166, 167], on-site exchange interactions [80] or superexchange interactions [168]. Moreover, one can use such realizations as quantum simulators to mimic specific condensed matter models. • Disordered systems: Interplay localization-interactions. Disorder plays a central role in condensed matter physics, and its presence leads to various novel types of effects and phenomena. One of the most prominent quantum signatures of disorder is Anderson localization [169] of the wave function of single particles in a random potential. In cold gases, controlled disorder, or pseudo– disorder might be created in atomic traps, or optical lattices by adding an optical potential created by speckle radiation, or by superposing several lattices with incommensurate periods of spatial oscillations [170, 171]. Other proposed methods are the admixture of different atomic species randomly trapped in sites distributed across the

6 sample and acting as impurities [172], or the use of an inhomogeneous magnetic field which modifies randomly, close to a Feshbach resonance, the scattering length of the atoms [173]. In fact, recently, experimental realization of Anderson localization of matter waves has been reported in a noninteracting BEC of 39 K in a pseudo-random potential [174], and in a small condensate of 87 Rb in a truly random potential in the course of expansion in a one-dimensional waveguide [175]. Three dimensional Anderson localization has also been reported for a spin-polarized atomic Fermi gas of 40 K [176] and for 87 Rb ultracold atoms [177]. The experiment reported in [175], as well as the appearance of the effective mobility edge due to the finite correlation length of the speckle induced disorder, was precisely predicted in [178]. The interplay between disorder and interactions is a very active research area, also in ultracold gases. For attractive interactions, disorder might destroy the possibility of superfluid transition (”dirty” superconductors) . Weak repulsive interactions play a delocalizing role, whereas very strong ones lead to Mott type localization [179], and insulating behavior. In the intermediate situations there exist a possibility of delocalized “metallic” phases. Cold atoms in optical lattices should allow to study the crossover between Anderson-like (Anderson glass) to Mott type (Bose glass) localization. First experimental signatures of a Bose glass [48, 180, 181], of the Anderson glass crossover [182] and of glassy behaviour in binary atomic mixtures [183] have been reported. Theoretical predictions [184– 187] indicate signatures of Anderson localisation in the presence of weak nonlinear interactions and quasi-disorder in BEC. One expects in such systems the appearance of a novel Lifshits glass phase [188], where bosons condense in a finite number of states from the low energy tail of the single particle spectrum. Very recently, the emergence of a disorder-induced insulating state [189] and the observation of many-body localization [190] in interacting fermions have been reported. • Disordered systems: spin glasses. Since the seminal papers of Edwards and Anderson [191], and Sherrington and Kirkpatrick [192] the question about the nature of the spin glass ordering has attracted a lot of attention [193–195]. The two competing pictures: the replica symmetry breaking picture of G. Parisi, and the droplet model of D.S. Fisher and D.A. Huse are probably applicable in some situations, and not applicable in others. Ultracold atoms in optical lattices might contribute to resolve this controversy by, for instance, studying independent copies with the same disorder, the so– called replicas. A measurement scheme for the determination of the disorder-induced correlation function between the atoms of two independent

replicas with the same disorder has been proposed [196]. Cold atomic physics might also add understanding of some quantum aspects, like for instance behavior of Ising spin glasses in transverse fields —i.e. in a truly quantum mechanical situation [197, 198]. More recently, the realization of the Dicke model with ultracold atoms in a single mode cavity [101], stimulated recently intensive studies of the connection between multi-mode Dicke models with random couplings and the spin glass physics [199, 200]. Also, the possibility of addressing NP versions of the number partitioning were mentioned and investigated in this context [201, 202]. • Spin glasses and D-Wave computers. In fact spin models are paradigms of multidisciplinary science. They are most relevant for various fields of physics, reaching from condensed matter to high energy physics, but they also find several applications beyond the physical sciences. In neuroscience, brain functions are modeled by interacting spin systems, going back to the famous Hopfield model of associative memory [203]. This directly relates to computer and information sciences, where pattern recognition or error-free coding can be achieved using spin models [204]. Importantly, many optimization problems, like number partitioning or the famous traveling salesman problem, belonging to the class of NP-hard problems, can be mapped onto the problem of finding the ground state of a specific spin model [205, 206]. This implies that solving a spin model itself is a task for which no general efficient classical algorithm is known to exist. A controversial development, supposed to provide also an exact numerical understanding of spin glasses, regards the D-Wave machine. These devices employing arrays of superconducting junctions, were recently introduced on the market. What they do is in fact that they solve (i.e. find the ground state of) classical spin glass models by using the, so called quantum annealing approach. Within this approach, if the system is trapped in a local minimum of energy, it can leave this configuration via quantum tunneling. Many researchers agree that D-Wave computers are genuine quantum simulators, because despite decoherence and coupling to a thermal bath, they are consistent with performing quantum annealing [207, 208], and with open quantum system dynamics [209], but the underlying mechanisms are not clear, and it remains an open question whether these machines provides a speed-up advantage over the best classical algorithms [210– 212]. All this triggers interest in alternative quantum systems, in particular quantum optical systems, designed to solve general spin models via quantum simulation. A noteworthy system for this goal are trapped ions: Nowadays, spin sys-

7 tems of trapped ions are available in many laboratories [65–68, 70, 71], and adiabatic state preparation, similar to quantum annealing, is experimental state-of-art. • Disordered systems: Large effects by small disorder. There are many examples of such situations. In classical statistical physics a paradigm is the random field Ising model in 2D —that looses spontaneous magnetization at arbitrarily small disorder—. In quantum physics the paradigmatic example is Anderson localization, which occurs at arbitrarily small disorder in 1D, and should occur also at arbitrarily small disorder in 2D. Cold atomic physics may address these questions, and, in fact, much more — cf. [213], [214] and [215], where disorder breaks the continuous symmetry in a spin system, and thus allows for long range ordering—. • High T c superconductivity. Despite many years of research, opinions on the nature of high T c superconductivity still vary quite appreciably [216]. It is, however, quite established —cf. contribution of P.W. Anderson in [216]— that understanding of the 2D Hubbard model in the, so called, t − J limit [160, 217–219] for two component (spin 1/2) fermions provides at least part of the explanation. The simulation of these models are very hard and numerical results are also full of contradictions. Cold fermionic atoms with spin (or pseudospin) 1/2 in optical lattices might provide a quantum simulator to resolve these problems [220] —see also [221]—. First experiments with both ”spinless”, i.e. polarized, as well as spin 1/2 unpolarized ultracold fermions [47, 222] have been realized; particularly spectacular are the recent observations of a fermionic Mott insulator state [52, 53]. It is also worth noticing that Bose-Fermi mixtures in optical lattices have already been intensively studied [49, 50, 223]; these systems might exhibit superconductivity due to boson-mediated fermion-fermion interactions. Superexchange interactions , demonstrated very recently in the context of ultracold atoms in optical lattices [168], are believed also to play an important role in the context of high-temperature superconductivity [224]. Remarkably, antiferromagnetic correlations for Fermion has been demonstrated in [225] in dimerized lattices via radiofrequency band transfer and observed in momentum space via spin sensitive Bragg scattering of light [226] and very recently also in situ via quantum gas microscopy [227–230]. Very recently antiferromagnetic • BCS-BEC crossover. Physics of high T c superconductivity can be also addressed with trapped ultracold gases. Weakly attracting spin-1/2 fermions in such situations undergo at (very) low temperatures the BCS Bardeen-Cooper-Schrieffer (BCS)

transition to a superfluid state of loosely bounded Cooper pairs. Weakly repulsive fermions, on the other hand may form bosonic molecules, which in turn may form at very low temperatures a BEC . Strongly interacting fermions undergo also a transition to the superfluid state, but at much higher T . Several groups have employed the technique of Feshbach resonances [231–233] —for a recent review on this technique see [234]— to observe such BCS-BEC crossover. For the recent status of experiments of the spin-balanced case see references in [14] and [13]—. There is much more controversy regarding imbalanced spin mixtures [235, 236]. First experimental signatures supporting pairing with finite momentum in spin-imbalance mixtures, the so-called FFLO state [237, 238], have been observed in one-dimensional Fermi gases [86]. Interestingly, topological nontrivial flat bands have been proposed for increasing the critical temperature of the superconducting transition [239]. • Frustrated antiferromagnets and spin liquids. The “rule of thumb” says that everywhere, in a vicinity of a high T c superconducting phase, there exists a (frustrated) antiferromagnetic phase. Frustrated antiferromagnets have been thus in the center of interest in condensed matter physics for decades. Particularly challenging here is the possibility of creating novel, exotic quantum phases, such as valence bond solids, resonating valence bond states, and various kinds of quantum spin liquids —spin liquids of I and II kind , according to C. Lhuillier [240, 241], and topological and critical spin liquids, according to M. P. A. Fisher [242, 243]—. Cold atoms offer also in this respect opportunities to create various frustrated spin models in triangular, or even kagomé lattices [164]. In the latter case, it has been proposed by Damski et al. [244, 245] that cold dipolar Fermi gases, or Bose-Fermi mixtures might allow to realize a novel state of quantum matter: quantum spin liquid crystal, characterized by Néel like order at low T —see also [246]—, accompanied by extravagantly high, liquid-like density of low energy excited states. • Topological order and quantum computation. Several very “exotic” spin systems with topological order have been proposed recently [247, 248] as candidates for robust quantum computing (for a recent review see [249]). Despite their unusual form, some of these models can be realized with cold atoms [163, 250]. Particularly interesting [251] is the recent proposal by Micheli et al. [250], who propose to use hetero-nuclear polar molecules in a lattice, excite them using microwaves to the lowest rotational level, and employ strong dipole-dipole interactions in the resulting spin model. The method provides an universal “toolbox” for spin models with designable range and spatial anisotropy of

8 couplings. Experimental achievements in cooling of heteronuclear molecules that may have a large electric dipole moment [49, 252] have opened possibilities in this direction. Recently, a gas of ultracold ground state Potassium-Rubidium molecules has been realized [253]. • Systems with higher spins. Lattice Hubbard models, or spin systems with higher spins are also related to many open challenges; perhaps the most famous being the Haldane conjecture concerning the existence of a gap, or its lack for the 1D antiferromagnetic spin chains with integer or halfinteger spins, respectively. Ultracold spinor gases [254, 255] might help to study these questions. Again, particularly interesting are in this context spinor gases in optical lattices [256–259], where in the strongly interacting limit the Hamiltonian reduces to a generalized Heisenberg Hamiltonian. Spinor condensates in optical lattices have been addressed experimentally for ferromagnetic [260– 263] and antiferromagnetic [264] interactions. Using Feshbach resonances [234] and varying the lattice geometry one should be able in such systems to generate a variety of regimes and quantum phases, including the most interesting antiferromagnetic (AF) regime. García-Ripoll et al. [165] proposed to use a duality between the antiferromagnetic (AF) and ferromagnetic (F) Hamiltonians, HAF = −HF , which implies that minimal energy states of HAF are maximal energy states of HF , and vice versa. Since dissipation and decoherence are practically negligible in such systems, and affect equally both ends of the spectrum, one can study AF physics with HF , preparing adiabatically AF states of interest. Ytterbium fermionic atoms with N different spin states in an optical lattices implementing the fermionic SU(N) Hubbard model have been recently experimentally investigated [265]. • Fractional quantum Hall states. Since the famous work of Laughlin [266], there has been enormous progress in our understanding of the fractional quantum Hall effect (FQHE) [267]. Nevertheless, many challenges remain open like the direct observation of the anyonic character of excitations or the observation of other kinds of strongly correlated states. FQHE states might be studied with trapped ultracold rotating gases [268, 269]. Rotation induces there effects equivalent to an “artificial” constant magnetic field directed along the rotation axis. There are proposals about how to detect directly fractional excitations in such systems [270]. Optical lattices might help in this task in two aspects. First, FQHE states of small systems of atoms could be observed in a lattice with rotating site potentials, or an array of rotating microtraps —cf. [271], [272],[273] and [274] and references therein—. Second, ”artificial” magnetic field

might be directly created in lattices via appropriate control of the tunneling (hopping) matrix element in the corresponding Hubbard model [275]. Such systems will also allow to create FQHE type states [276–279]. Klein and Jaksch [280] have recently proposed to immerse a lattice gas in a rotating Bose condensate; tunneling in the lattice becomes then partially mediated by the phonon excitations of the BEC and mimics the artificial magnetic field effects. Last, but not least, a direct approach employing lattice rotation has been developed both in theory [281, 282], and in experiments [283]. The recent wave of very successful experiments creating artificial or synthetic magnetic fields employing laser induced gauge fields [81, 82] that are achieved by using spatial dependent optical coupling between different internal states of atoms. Such approach is free from rotational restrictions. Very recently, this successful wave have invested also lattice experiments. Paradigmatic 2D models displaying a topological insulating behavior like the Hofstadter [284] and the Haldane [285] models have been realized experimentally in shallow harmonic traps in [286, 287] and [288], respectively, by exploiting superlattices and Bragg pulses. The Hofstadter model has been also realized in ladders and slabs, that to say in lattices with sharp boundaries in one narrow dimension. Such dimension can be real as in the experiment by [289] or synthetic [290], that to say with the rungs of the ladder formed by the internal spin states of atoms, which provide an extra dimension to the system [291]. The synthetic Hofstadter slabs has been realized experimentally for bosons [292] and fermions [293] and the edge currents observed for the first time via spin-dependent measurements. Real and synthetic Hofstadter ladders and slabs offer a very promising route to the observation of many-body quantum Hall physics, both for bosons [294, 295] and fermions [296, 297]. • Lattice gauge fields. Gauge theories, and in particular lattice gauge theories (LGT) [298] are fundamental for both high energy physics and condensed matter physics, and despite the progress of our understanding of LGT, many questions in this area remain open (see e.g. [299]). Physics of cold atoms might help here in two aspects: “artificial” non Abelian magnetic fields may be created in lattice gases via appropriate control of the hopping matrix elements [300] or in trapped gases using effects of electromagnetically induced transparency [301]. One of the most challenging tasks in this context concerns the possibility of realizing generalizations of Laughlin states with possibly non Abelian fractional excitations. Another challenge concerns the possibility of “mimicking” the dynamics of gauge fields. In fact, dynamical realizations of U(1) Abelian gauge theory, that in-

9 volve ring exchange interaction in a square lattice [302], or 3 particle interactions in a triangular lattice [303, 304] have been also proposed. In the last four years the effort of simulating LGT in optical lattices has received new impulse. Most of the attention has focused on gauge magnets or link models [305–307], which can be viewed as spin version of ordinary Hamiltonian LGT and, as such, are easier to be simulated. Both Abelian and non-Abelian LGT can be simulated by exploiting angular momentum conservation [308, 309], SU(N)-invariant interaction in earth-alkali like atoms [310, 311], or the long-range interaction induced by Rydberg atoms [312, 313]. The proposed simulators would be capable to probe the confinement of charges, for instance, by measuring the string tension. Similar simulators have been proposed also for superconducting qubit [314] and trapped ions [315], where string breaking in Schwinger model has been very recently experimentally demonstrated with four ions [77]. These developments in quantum simulation has triggered also parallel developments in classical simulation both in 1D [316–318] and 2D or more [319, 320].

nant dipole interactions. More recently, also dipolar gases of Dysprosium [333] and Erbium [334] have been cooled up to reach quantum degeneracy. Very recently, dipolar gases [335] and polar molecules [336] have been used to simulate quantum magnetism. Dipolar BECs and BCS states of trapped gases are expected to exhibit very interesting dependence on the trap geometry. Dipolar ultracold gases in optical lattices, described by extended Hubbard models, should allow to realize various quantum insulating “solid” phases, such as phase checkerboard (CB), and superfluid phases such as supersolid (SS) phase [337–339]. Particularly interesting in this context are the rotating dipolar gases (RDG) . Bose-Einstein condensates of RDGs exhibit novel forms of vortex lattices: square, “stripe crystal”, and “bubble crystal” lattices [340]. [341] have demonstrated that the pseudo-hole gap survives the large N limit for the Fermi RDGs, making them perfect candidates to achieve the strongly correlated regime, and to realize Laughlin liquid at filling ν = 1/3, and quantum Wigner crystal at ν ≤ 1/7 [342] with mesoscopic number of atoms N ≃ 50 − 100.

• Superchemistry. This is a challenge of quantum chemistry, rather than condensed matter physics: to perform a chemical reaction in a controlled way, by using photoassociation or Feshbach resonances from a desired initial state to a desired final quantum state. Jaksch et al. [321] proposed to use a Mott insulator (MI) with two identical atoms, to create via photoassociation, first a MI of homonuclear molecules , and then a molecular SF via “quantum melting” . A similar idea was applied to heteronuclear molecules [322] in order to achieve molecular SF. Bloch’s group have indeed observed photoassociation of 87 Rb molecules in a MI with two atoms per site [323], while Rempe’s group has realized the first molecular MI using Feshbach resonances [54]. Formation of three-body Efimov trimer states was observed in trapped Cs atoms by Grimm’s group [324]. This process could be even more efficient in optical lattices [325]. An overview of the subject of cold chemistry can be found in [326], and in particular about cold Feshbach molecules in [327]. Control and creation of deeply bound molecules in the presence of an optical lattice has been reported in [328].

• Wigner crystals or self-assembled lattices. Wigner, or Coulomb type of crystals are predicted to be formed due to long range repulsive dipolar atomatom or molecule-molecule interactions [343–345], or ion-ion Coulomb interactions in the absence of a lattice [64].

• Ultracold dipolar gases. Some of the most fascinating experimental and theoretical challenges of the modern atomic and molecular physics concern ultracold dipolar quantum gases —for reviews, see [329],[330] and [331]—. The recent experimental realization of the dipolar Bose gas of Chromium [332], and the progress in trapping and cooling of dipolar molecules [253] have opened the path towards ultracold quantum gases with domi-

Many of the above mentioned challenges are discussed discussed in the subsequent chapters of our book [1]. The interested reader should, however, extend her/his knowledge by turning toward the excellent books of Pitaevskii and Stringari [9] and Pethick and Smith [346] on Bose-Einstein Condensation in the weakly interacting regime, or to the books of Fetter and Walecka [347] or X.G. Wen [348] for many body and quantum field theory, among others. IV.

CONCLUSIONS

EU as well many other countries have concentrated recently considerable efforts to support quantum information science and technology. EU in particular will launch a Quantum Flagship, which will seek for technological advances in four pillars of the QI science; Quantum Computing and Simulations, Quantum Communications, Quantum Sensing and Metrology, and Quantum information Theory and Software. While universal quantum computers remains still very challenging, quantum simulators already exist in the labs. Many of those are analogue or digital simulators of “interesting” quantum phenomena. In the recent year, however, a lot of effort has been devoted to quantum annealers, like D- wave machines that are designed

10 to solve classical NP-complete or at least ultra-complex problems [202, 349, 350]. Quantum annealers are perhaps the first quantum computing devices that have the chance to enter real technology and change our everyday life. In fact, a decisive progress toward definitive success of quantum computing devices may be achieved by integrating the different platforms [351] that are best suited for different tasks. The development of such integrated structure would incarnate fully the spirit of the Quantum Flagship, and has quantum optics in its heart. Ones of the most intriguing applications of quantum simulators are nowadays facing towards problems that are not restricted to condensed matter, and even to physics in stricto sensu [352]. Among them, simulation of Lattice Gauge Theories is especially fascinating and may lead to breaking the 20th century distinction between low and high energy physics. Such simulators may serve also for testing holographic principle [353]. In fact, quantum simulation of high energy phenomena and gravitation [354–356] is not the only way in which quantum optics and quantum technologies may have a revolutionary impact over our understanding of Nature at all scales. Indeed, in addition to the celebrate success of the interferometer LIGO in detecting gravitational waves [357]

as predicted by Einstein about a century ago [358, 359], the impressive progresses in the optical clock standards [360] are expected to lead in the next decade to direct tests of fundamental laws of physics by proving tighter and tighter bounds on the stability of fundamental constants [361]. Thus, exciting time and challenges are expecting the quantum optics community in the following years that require a common effort.

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Quantum workers of the world, unite! This is not the final struggle, but... Let us group together, and tomorrow The Quantum Internationale Will be the human race. We acknowledge financial support from John Templeton Foundation, European Union (EU IP SIQS, EU Grant PROACT QUIC, EU STREP EQUAM), the European Research Council (ERC AdG OSYRIS), and the Spanish MINECO (National Plan projects FOQUS – FIS201346768 and Q-TRIP – FIS2014-57460-P) and the Generalitat de Catalunya AGAUR (2014 SGR 874 and SGR20141639).

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