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I. COLD ATOMS FROM A HISTORICAL QUANTUM awarded the Prize to S. Chu [25], C. Cohen- OPTICAL PERSPECTIVE Tannoudji [26] and W.D. Phillips [27] “for the de- velopment of methods to cool and trap atoms with The third quantum revolution does not arrive com- laser light”. Laser cooling and mechanical manip- pletely unexpected: the developments of atomic physics ulations of particles with light [28] was essential and quantum optics in the last 30 years have inevitably for development of completely new areas of atomic led to it. In the seventies and eighties of the XX cen- physics and quantum optics, such as atom optics tury, atomic physics was a very well established and re- [29], and for reaching new territories of precision spectful area of physics. It was, however, by no means metrology and quantum engineering. a “hot” area. On the theory side, even though one had • Laser cooling combined with evaporation cool- to deal with complex problems of many electron sys- ing allowed, in 1995, the experimental observation tems, most of the methods and techniques were already Bose-Einstein condensation (BEC) [2–4], a phe- developed. The major problems that were considered nomenon predicted by S. Bose and A. Einstein concerned mainly questions related on optimisation of more than 70 years earlier. As we said before, these these methods. These questions were reflecting an evo- experiments marked evidently the birth of a new lutionary progress, rather than a revolutionary search era. E.A. Cornell and C.E. Wieman [7] and W. Ket- for totally new phenomena. On the experimental side, terle [8] received the Nobel Prize in 2001, “for the quantum optics at that time was entering its Golden Age. achievement of Bose-Einstein condensation in di- The development of laser physics and nonlinear optics lute gases of alkali atoms, and for early fundamen- led in 1981 to the Nobel prize for A.L. Schawlow and tal studies of the properties of the condensates”. N. Bloembergen “for their contribution to the develop- This was truly a breakthrough moment, in which ment of laser spectroscopy”. On the other hand, stud- “atomic physics and quantum optics has met con- ies of quantum systems at the single particle level cul- densed matter physics” [8]. Condensed matter minated in 1989 with the Nobel prize for H.G. Dehmelt community at that time remained, however, still and W. Paul “for the development of the ion trap tech- reserved. After all, BEC was observed in weakly nique”, shared with N.F. Ramsey ”for the invention of interacting dilute gases, where it is very well de- the separated oscillatory fields method and its use in the scribed by the mean field Bogoliubov-de Gennes hydrogen maser and other atomic clocks”. theory [9], developed for homogeneous systems in Theoretical quantum optics was born in the 60’ties the 50’ties. with the works on quantum coherence theory devel- oped the 2005 Noble prize winner, R.J. Glauber [21, 22], • The study of quantum correlations and entangle- and with the development of the laser theory in the late ment. The seminal theoretical works of the late A. 60’ties by H. Haken and by M.O. Scully and W.E. Lamb Peres [30, 31], the proposals of quantum cryptog- (Nobel laureate of 1955). In the 70’ties and 80’ties, how- raphy by C. H. Bennett and G. Brassard [32], and ever, theoretical quantum optics was not considered to A. K. Ekert [33], the quantum communication pro- be a separate, established area of theoretical physics. posals by C. H. Bennett and S. J. Wiesner [34] and One of the reasons of this state of the art, was that in- C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, deed quantum optics at that time was primarily deal- A. Peres, and W. K. Wootters [35], the discovery of ing with single particle problems. Most of the many the quantum factorizing algorithm by P. Shor [36], body problems of quantum optics, such as laser theory, and the quantum computer proposal by J. I. Cirac or more generally optical instabilities [23], could have and P. Zoller [37] gave birth to the field of quantum been solved either using linear models, or employing information [38, 39] (for the latest version of the relatively simple versions of the mean field approach. quantum information road map see [40]). These Perhaps the most sophisticated theoretical contributions studies, together with the rapid development of concerned understanding of quantum fluctuations and the theory have led to an enormous progress in our quantum noise [23, 24]. understanding of quantum correlations and entan- This situation has drastically changed due to the un- glement and, in particular, how to prepare and use precedented level of quantum engineering, i.e. prepara- entangled states as a resource. The impulses from tion, manipulation, control and detection of quantum quantum information enter nowadays constantly systems developed and achieved by atomic physics and into the physics of cold atoms, molecules and ions, quantum optics. There are several seminal discoveries and stimulate new approaches. It is very probable that have triggered these changes: that the first quantum computers will be, as sug- gested already by Feynman [41, 42], computers of • The cooling and trapping of atoms, ions and special purpose – quantum simulators (QS) [43], molecules have reached regimes of very low tem- that will efficiently simulate quantum many body peratures (today down to nano-Kelvin!) and pre- systems that otherwise cannot be simulated using cision that were considered unattainable just two “classical” computers [39]. decades ago. These developments were recog- nized by the Nobel Foundation in 1997, who • Optical lattices and Feshbach resonances. The 3

physics of ultracold atoms entered the area of could look at phonons in ion self-assembled crys- strongly correlated systems with the seminal tals; these are also predicted to exhibit interest- proposal of Jaksch et al. [15] on how to achieve ing collective behaviour from Bose condensation the transition superfluid-Mott insulator transition to strongly correlated states [72, 73]. Trapped ions in cold atoms using optical lattices. The proposal may be used also to simulate mesoscopic spin- was based on the bosonic Mott insulator transi- boson models [74]. Combined with optical traps tion proposed in condensed matter [44], but the or ion microtrap array methods can be used to sim- authors of this paper were in fact motivated by ulate a whole variety of spin models with tunable the possibility of realizing quantum computing interactions in a wide range of spatial dimensions with cold atoms in a lattice. Transition to the Mott and geometries [75]. Finally, they are promising insulator state was supposed to be an efficient candidates for the realization of many-body inter- way of preparing a quantum register with a fixed actions [76]. All these theoretical proposals and the number of atoms per lattice site. Entanglement spectacular progress in the experimental trapped between atoms could be achieved via controlled ion community pushes trapped ions physics to be collisions [45]. The experimental observation soon used as widely as cold atoms to mimic con- of the superfluid-Mott insulator transition by densed matter physics and beyond, as for instance the Bloch–H´’ansch group [16] was without any in [77]. doubts a benchmark in the studies of strongly correlated systems with ultracold atoms [46]. Several other groups have observed bosonic II. QUANTUM SIMULATORS superfluid-Mott insulator transitions in pure Bose systems [47], in disordered Bose systems [48], in Before we proceed we would like to make a short de- Bose-Fermi [49, 50], and Bose-Bose mixtures [51]. tour to explain in a little more detail the concept of quan- The possibility to change the collision properties of tum simulators. XX century was the age of information ultracold atoms by means of tuning the Feschbach and computers. But, even supercomputers have their resonances of the atomic species has been another limitations – as we well know for instance their ability tool of inestimable value. Such a tool has lead to to predict weather, a paradigmatic classical chaotic phe- the fermionic Mott insulator [52, 53]. Also, Mott nomenon, is very restricted. For this reason already in insulator state of molecules have been created classical computer science the concept of special purpose [54], as well as bound repulsive pairs of atoms (i.e. computers was developed. These “classical simulators” pairs of atoms at a site that cannot release their are not like universal classical computers – they can only repulsive energy due to the band structure of the simulate or calculate certain restricted class of models spectrum in the lattice) have been observed [55]. describing Nature. The best simulations of classical dis- ordered systems, such as spin glasses, are nowadays ob- tained with such computers of special purpose – “classi- • Cold trapped ions were initially investigated to re- cal simulators”. alize the Cirac-Zoller quantum gate [37], and to at- The situation is even more dramatic in quantum tempt to build an scalable quantum computer us- physics and chemistry. We know very well that an uni- ing a bottom-up approach [56, 57]. Recent progress versal quantum computer will revolutionize our tech- in this research line can be found in [58]. A new nology in the future. Unfortunately, this future does stream of ideas using cold ions for quantum sim- not seem to be very close at hand, and definitely con- ulators have recently appeared. First proposals cerns tens of years. However, quantum computers with have shown that ion-ion interactions mediated by a special purpose, i.e. quantum simulators, exist since phonons can be manipulated to implement vari- already 5 years (cf. [78]). ous spin models [59–61], where the spin states cor- Various quantum phenomena such as high-Tc super- respond to internal states of the ion. In tight lin- conductivity or quark confinement are still awaiting uni- ear traps, one could realize the spin chains with versally accepted explanations because of the compu- interactions decaying as (distance)−3, i.e. as in tational complexity of solving the simplified theoret- the case of dipole-dipole interactions. Such spin ical models designed to capture the relevant physics. chains may serve as quantum neural network mod- Richard Feynman suggested in 1982 [41, 42], solving els and may be used for adiabatic quantum infor- such models by ”quantum simulation”. He pointed out mation processing [62, 63]. More interestingly, ions that it might be possible sometimes to find a simpler can be employed as quantum simulators, both in and experimentally more accessible system to mimic the 1D and in 2D, where the ions form self-assembled quantum system of interest. Feynman’s idea was moti- Coulomb crystals [64]. First steps towards the vated by the complexity of classical simulations of quan- experimental realization of these ideas has been tum systems. Suppose we want to study a system of reported [65]. These results have been recently N spins 1/2; then the dimension of the Hilbert space is extended in [66–71] Alternatively to spins, one 2N, and the number of coefficients we need to describe 4 the wave function of the system may be in principle just it is desirable to realize quantum simulators to as large. As N grows, this number quickly becomes simulate and to observe novel, hitherto only the- larger than the number of atoms in the universe, so clas- oretically predicted phenomena, even though it sical simulations are evidently impossible. Of course, might be possible to simulate these phenomena ef- in practice this number may be very much reduced by ficiently with present-day computers. Simulating using clever representations of the wave functions, but and observing is more than just simulating. in general there are many quantum systems, which are very hard to simulate classically. The modern concept • A quantum simulator should allow for broad con- of quantum simulators is not exactly the same as that of trol of the parameters of the simulated model, and Feynman. Since condensed matter systems are typically for control of the preparation, manipulation and very complex, theorists construct simplified models. In detection of the states of the system. In particular, this simplification, symmetries of the original problem it is important to be able to set the parameters in are kept intact, and the concept of universality is used: such a way that the model becomes tractable using different Hamiltonians with similar symmetry proper- classical simulations. This provides the possibility ties belong to the same universality classes, i.e. they of validating the quantum simulator. exhibit the same phase transitions and have the same Fueled by the prospect of solving a broad range of critical exponents. Unfortunately, even these simplified long-standing problems in strongly-correlated systems, models are often difficult to understand. A paradigmatic the tools to design, build, and implement QSs [1, 41, 79] example of such a situation concerns high-T supercon- c have rapidly developed and are now reaching very so- ductivity of cuprates, where it is believed that the basic phisticated levels [78]. There exist many proposals and physics is captured by an array of weakly coupled 2D already many realizations of quantum simulators em- Hubbard models for electrons, i.e. spin-1/2 fermions. ploying ultracold atoms and molecules in traps and in Even this simple model reduced to one 2D plane does optical lattices (cf. [80–91], for a review see also [14]), not allow for accurate treatment, and there is much con- probing quantum dynamics (cf. [92, 93]), employing ul- troversy concerning, for instance, the phase diagram or tracold trapped ions (cf. [65, 67, 94–98], for recent re- the character of the transitions. The role of quantum views see [99, 100]), atoms in single or in arrays of cav- simulators, as proposed in many quantum information ities (cf. [101]), ultracold atoms/ions in arrays of traps, projects, will be to simulate simple models, such as 2D ultracold atoms near nano-structures and multidimen- Hubbard models, and obtain a better understanding of sional plasmonic traps [102–105], photons [106–110], ar- them, rather than try to simulate the full complexity of rays of quantum dots, circuit quantum electrodynamics the condensed matter. (QED) and polaritons (cf. [111–115], for a recent review A “working” definition of a quantum simulator could see [116]), artificial lattices in solid state [117], nuclear be as follows [1, 79]: magnetic resonance (NMR) systems [118–121] and su- • A quantum simulator is an experimental system perconducting qubits. that mimics a simple model, or a family of sim- In addition to our book [1] there exist excellent recent ple models, of condensed matter (or high-energy reviews, in particular in the focus issue of Nature Physics physics, or quantum chemistry...). Insight [122–127]. At the current pace, it is expected that we will soon reach the ability to finely control many- • The simulated models have to be of some rele- body systems whose description is outside the reach of vance for applications and/or our understanding a classical computer. For example, modeling interest- of the challenges of the above-mentioned areas ing physics associated with a quantum system involving of physics, and of the new challenges in Physics, 50-100 spin-1/2 particles – whose general description re- 50 100 15 30 Chemistry and Biology that are still to be identi- quires 2 − 2 ≈ 10 − 10 amplitudes – is out of the fied. reach of current classical supercomputers, but perhaps within the grasp of a quantum simulators. • The simulated models should be computationally Obviously, the real-world implementations of a quan- intractable for classical computers. Note that this tum simulation will always face experimental imperfec- statement may have two meanings: i) an efficient tions, such as noise due to finite precision instruments (scalable to large system size) algorithm to sim- and interactions with the environment. The quantum ulate the model might not exist, or might not be simulator – as envisioned by Feynman – is fundamen- known; ii) the efficient scalable algorithm may be tally an analog device, in the sense that all operations known, but the size of the simulated model is too are carried out continuously. However, errors in an ana- large to be simulated under reasonable time and log device (also continuous, like temperature in the ini- memory restrictions. The latter situation, in fact, tial state, or the signal-to-noise ratio of measurement) occurs with classical simulations of the Bose or can propagate and multiply uncontrollably [128]. In- Fermi Hubbard models as compared to their ex- deed, Landauer, a father of the studies of the physics perimental quantum simulators. There might also of information, questioned whether quantum coherence be exceptions to the general rule. For instance, was truly a powerful resource for computation because 5 it required a continuum of possible superposition states a 1D Luttinger liquid of cold bosonic atoms ac- that were “analog” in nature [129]. In contrast to digital complishes a spin-boson model with Ohmic cou- quantum simulators, where the error correction schemes pling, which exhibits a dissipative phase transition might be in principle applied, the question of validation, and allows to directly measure atomic Luttinger calibration and reliability of analog quantum simulators parameters. Also, it has been shown that in the is very subtle and very ”hot” [130]. 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 III. COLD ATOMS AND QUANTUM OPTICS AT THE bath [150] FRONTIERS OF PHYSICS • 2D systems. According to the celebrated Mermin- Quantum optics and physics of cold atoms touches Wagner-Hohenberg theorem, 2D systems with nowadays, among others, common frontiers of contem- continuous symmetry do not exhibit long range porary physics with condensed matter physics, quan- order at temperatures T > 0. 2D systems may, tum many body physics, nuclear physics, quantum field however, undergo Kosterlitz-Thouless-Berezinskii theory, high energy physics, and even astrophysics. In transition (KTB) to a state in which the correlations particular, many important challenges of the latter disci- decay is algebraic, rather than exponential. Al- plines can be addressed within cold atoms and quantum though KTB transition has been observed in liquid optical wizardry: Helium [151, 152], its microscopic nature (binding of vortex pairs) has never been seen. Recent exper- • 1D systems. The role of quantum fluctuations iments [153–159] make an important step in this and correlations is particularly important in 1D. direction. The theory of 1D systems is very well developed due to the existence of exact methods such as • Hubbard and spin models. Many very important ex- Bethe Ansatz and quantum inverse scattering the- amples of strongly correlated states in condensed ory (cf. [131]), powerful approximate approaches, matter physics are realized in various types of such as bosonization, or conformal field theory Hubbard models [131, 160]. While Hubbard mod- —cf. [132]—, and efficient computational meth- els in condensed matter are “reasonable carica- ods, such as density matrix Renormalization group tures” of real systems, ultracold atomic gases in op- technique (DMRG) —cf. [133]—. There are, how- tical lattices allow to achieve practically perfect re- ever, many open experimental challenges that have alizations of a whole variety of Hubbard models not been so far directly and clearly realized in con- [161]. Similarly, in certain limits Hubbard models densed matter, and can be addressed with cold reduce to various spin models; again cold atoms atoms —for a review see [134, 135]—. Exam- and ions allow for practically perfect realizations ples include atomic Fermi, or Bose analogues of of such spin models —see for instance [162–165] spin-charge separation, or more generally observa- and [66, 68, 70, 71, 87]—. Specifically, spin-spin in- tions of microscopic properties of Luttinger liquids teractions between neighboring atoms can be im- [136–138]. Experimental observations of the 1D plemented by bringing the atoms together on a gas in the deep Tonks-Girardeau regime by Pare- single site and carrying out controlled collisions des et al. [139] —see also [140],[141],[47] and [45, 166, 167], on-site exchange interactions [80] or [142]— were the first steps in this direction. The superexchange interactions [168]. Moreover, one recent achievement of Tonks regime in 2008 em- can use such realizations as quantum simulators ploying dissipative processes (two body losses) is to mimic specific condensed matter models. perhaps the first experimental example of how to • control a system by making use of its coupling Disordered systems: Interplay localization-interactions. to the environment [143–146]. See also the recent Disorder plays a central role in condensed mat- progress towards deep Tonks-Girardeau regime in ter physics, and its presence leads to various [147]. novel types of effects and phenomena. One of the most prominent quantum signatures of dis- • Spin-boson model. A two-level system coupled to a order is Anderson localization [169] of the wave bosonic reservoir is a paradigmatic model both, in function of single particles in a random poten- quantum dissipation theory in quantum optics and tial. In cold gases, controlled disorder, or pseudo– as quantum phase transition in condensed matter disorder might be created in atomic traps, or op- where it is termed as the spin-boson model —for a tical lattices by adding an optical potential cre- review see [148]—. It has also been proposed [149] ated by speckle radiation, or by superposing sev- that an atomic quantum dot, i.e., a single atom in eral lattices with incommensurate periods of spa- a tight optical trap coupled to a superfluid reser- tial oscillations [170, 171]. Other proposed meth- voir via laser transitions, may realize this model. ods are the admixture of different atomic species In particular, atomic quantum dots embedded in randomly trapped in sites distributed across the 6

sample and acting as impurities [172], or the use replicas with the same disorder has been proposed of an inhomogeneous magnetic field which modi- [196]. Cold atomic physics might also add un- fies randomly, close to a Feshbach resonance, the derstanding of some quantum aspects, like for in- scattering length of the atoms [173]. In fact, re- stance behavior of Ising spin glasses in transverse cently, experimental realization of Anderson local- fields —i.e. in a truly quantum mechanical sit- ization of matter waves has been reported in a non- uation [197, 198]. More recently, the realization interacting BEC of 39K in a pseudo-random poten- of the Dicke model with ultracold atoms in a sin- tial [174], and in a small condensate of 87Rb in a gle mode cavity [101], stimulated recently inten- truly random potential in the course of expansion sive studies of the connection between multi-mode in a one-dimensional waveguide [175]. Three di- Dicke models with random couplings and the spin mensional Anderson localization has also been re- glass physics [199, 200]. Also, the possibility of ported for a spin-polarized atomic Fermi gas of 40K addressing NP versions of the number partition- [176] and for 87Rb ultracold atoms [177]. The ex- ing were mentioned and investigated in this con- periment reported in [175], as well as the appear- text [201, 202]. ance of the effective mobility edge due to the fi- • Spin glasses and D-Wave computers nite correlation length of the speckle induced dis- . In fact spin models are paradigms of multidisciplinary sci- order, was precisely predicted in [178]. The inter- play between disorder and interactions is a very ence. They are most relevant for various fields of physics, reaching from condensed matter to high active research area, also in ultracold gases. For energy physics, but they also find several appli- attractive interactions, disorder might destroy the possibility of superfluid transition (”dirty” super- cations beyond the physical sciences. In neuro- science, brain functions are modeled by interact- conductors) . Weak repulsive interactions play a delocalizing role, whereas very strong ones lead ing spin systems, going back to the famous Hop- field model of associative memory [203]. This to Mott type localization [179], and insulating be- havior. In the intermediate situations there ex- directly relates to computer and information sci- ences, where pattern recognition or error-free cod- ist a possibility of delocalized “metallic” phases. Cold atoms in optical lattices should allow to study ing can be achieved using spin models [204]. Im- portantly, many optimization problems, like num- the crossover between Anderson-like (Anderson glass) to Mott type (Bose glass) localization. First ber partitioning or the famous traveling salesman problem, belonging to the class of NP-hard prob- experimental signatures of a Bose glass [48, 180, 181], of the Anderson glass crossover [182] and of lems, can be mapped onto the problem of finding the ground state of a specific spin model [205, 206]. glassy behaviour in binary atomic mixtures [183] have been reported. Theoretical predictions [184– This implies that solving a spin model itself is a task for which no general efficient classical algo- 187] indicate signatures of Anderson localisation in the presence of weak nonlinear interactions and rithm is known to exist. A controversial develop- ment, supposed to provide also an exact numerical quasi-disorder in BEC. One expects in such sys- tems the appearance of a novel Lifshits glass phase understanding of spin glasses, regards the D-Wave machine. These devices employing arrays of su- [188], where bosons condense in a finite number of states from the low energy tail of the single par- perconducting junctions, were recently introduced on the market. What they do is in fact that they ticle spectrum. Very recently, the emergence of a solve (i.e. find the ground state of) classical spin disorder-induced insulating state [189] and the ob- quantum an- servation of many-body localization [190] in inter- glass models by using the, so called nealing approach. Within this approach, if the sys- acting fermions have been reported. tem is trapped in a local minimum of energy, it • Disordered systems: spin glasses. Since the seminal can leave this configuration via quantum tunnel- papers of Edwards and Anderson [191], and Sher- ing. Many researchers agree that D-Wave comput- rington and Kirkpatrick [192] the question about ers are genuine quantum simulators, because de- the nature of the spin glass ordering has attracted spite decoherence and coupling to a thermal bath, a lot of attention [193–195]. The two competing they are consistent with performing quantum an- pictures: the replica symmetry breaking picture nealing [207, 208], and with open quantum sys- of G. Parisi, and the droplet model of D.S. Fisher tem dynamics [209], but the underlying mecha- and D.A. Huse are probably applicable in some sit- nisms are not clear, and it remains an open ques- uations, and not applicable in others. Ultracold tion whether these machines provides a speed-up atoms in optical lattices might contribute to resolve advantage over the best classical algorithms [210– this controversy by, for instance, studying inde- 212]. All this triggers interest in alternative quan- pendent copies with the same disorder, the so– tum systems, in particular quantum optical sys- called replicas. A measurement scheme for the tems, designed to solve general spin models via determination of the disorder-induced correlation quantum simulation. A noteworthy system for function between the atoms of two independent this goal are trapped ions: Nowadays, spin sys- 7

tems of trapped ions are available in many labora- transition to a superfluid state of loosely bounded tories [65–68, 70, 71], and adiabatic state prepara- Cooper pairs. Weakly repulsive fermions, on the tion, similar to quantum annealing, is experimen- other hand may form bosonic molecules, which in tal state-of-art. turn may form at very low temperatures a BEC . Strongly interacting fermions undergo also a tran- • Disordered systems: Large effects by small disor- sition to the superfluid state, but at much higher der. There are many examples of such situa- T. Several groups have employed the technique tions. In classical statistical physics a paradigm of Feshbach resonances [231–233] —for a recent is the random field Ising model in 2D —that review on this technique see [234]— to observe looses spontaneous magnetization at arbitrarily such BCS-BEC crossover. For the recent status of small disorder—. In quantum physics the paradig- experiments of the spin-balanced case see refer- matic example is Anderson localization, which oc- ences in [14] and [13]—. There is much more con- curs at arbitrarily small disorder in 1D, and should troversy regarding imbalanced spin mixtures [235, occur also at arbitrarily small disorder in 2D. Cold 236]. First experimental signatures supporting atomic physics may address these questions, and, pairing with finite momentum in spin-imbalance in fact, much more — cf. [213],[214] and [215], mixtures, the so-called FFLO state [237, 238], have where disorder breaks the continuous symmetry been observed in one-dimensional Fermi gases in a spin system, and thus allows for long range [86]. Interestingly, topological nontrivial flat bands ordering—. have been proposed for increasing the critical tem- • High Tc superconductivity. Despite many years of perature of the superconducting transition [239]. research, opinions on the nature of high T su- c • perconductivity still vary quite appreciably [216]. Frustrated antiferromagnets and spin liquids. The It is, however, quite established —cf. contribu- “rule of thumb” says that everywhere, in a vicin- tion of P.W. Anderson in [216]— that understand- ity of a high Tc superconducting phase, there exists ing of the 2D Hubbard model in the, so called, a (frustrated) antiferromagnetic phase. Frustrated t − J limit [160, 217–219] for two component (spin antiferromagnets have been thus in the center of 1/2) fermions provides at least part of the ex- interest in condensed matter physics for decades. planation. The simulation of these models are Particularly challenging here is the possibility of very hard and numerical results are also full of creating novel, exotic quantum phases, such as va- contradictions. Cold fermionic atoms with spin lence bond solids, resonating valence bond states, (or pseudospin) 1/2 in optical lattices might pro- and various kinds of quantum spin liquids —spin vide a quantum simulator to resolve these prob- liquids of I and II kind , according to C. Lhuil- lems [220] —see also [221]—. First experiments lier [240, 241], and topological and critical spin liq- with both ”spinless”, i.e. polarized, as well as uids, according to M. P.A. Fisher [242, 243]—. Cold spin 1/2 unpolarized ultracold fermions [47, 222] atoms offer also in this respect opportunities to cre- have been realized; particularly spectacular are ate various frustrated spin models in triangular, or the recent observations of a fermionic Mott in- even kagomé lattices [164]. In the latter case, it has sulator state [52, 53]. It is also worth noticing been proposed by Damski et al.[244, 245] that cold that Bose-Fermi mixtures in optical lattices have al- dipolar Fermi gases, or Bose-Fermi mixtures might ready been intensively studied [49, 50, 223]; these allow to realize a novel state of quantum matter: systems might exhibit superconductivity due to quantum spin liquid crystal, characterized by Néel boson-mediated fermion-fermion interactions. Su- like order at low T —see also [246]—, accompa- perexchange interactions , demonstrated very re- nied by extravagantly high, liquid-like density of cently in the context of ultracold atoms in optical low energy excited states. lattices [168], are believed also to play an impor- • Topological order and quantum computation. Several tant role in the context of high-temperature super- very “exotic” spin systems with topological order conductivity [224]. Remarkably, antiferromagnetic have been proposed recently [247, 248] as candi- correlations for Fermion has been demonstrated in dates for robust quantum computing (for a recent [225] in dimerized lattices via radiofrequency band review see [249]). Despite their unusual form, transfer and observed in momentum space via spin some of these models can be realized with cold sensitive Bragg scattering of light [226] and very atoms [163, 250]. Particularly interesting [251] is recently also in situ via quantum gas microscopy the recent proposal by Micheli et al.[250], who pro- [227–230]. Very recently antiferromagnetic pose to use hetero-nuclear polar molecules in a lat- • BCS-BEC crossover. Physics of high Tc supercon- tice, excite them using microwaves to the lowest ro- ductivity can be also addressed with trapped ul- tational level, and employ strong dipole-dipole in- tracold gases. Weakly attracting spin-1/2 fermions teractions in the resulting spin model. The method in such situations undergo at (very) low temper- provides an universal “toolbox” for spin models atures the BCS Bardeen-Cooper-Schrieffer (BCS) with designable range and spatial anisotropy of 8

couplings. Experimental achievements in cooling might be directly created in lattices via appropri- of heteronuclear molecules that may have a large ate control of the tunneling (hopping) matrix ele- electric dipole moment [49, 252] have opened pos- ment in the corresponding Hubbard model [275]. sibilities in this direction. Recently, a gas of ultra- Such systems will also allow to create FQHE type cold ground state Potassium-Rubidium molecules states [276–279]. Klein and Jaksch [280] have re- has been realized [253]. cently proposed to immerse a lattice gas in a ro- tating Bose condensate; tunneling in the lattice be- • Systems with higher spins. Lattice Hubbard mod- comes then partially mediated by the phonon exci- els, or spin systems with higher spins are also re- tations of the BEC and mimics the artificial mag- lated to many open challenges; perhaps the most netic field effects. Last, but not least, a direct famous being the Haldane conjecture concerning approach employing lattice rotation has been de- the existence of a gap, or its lack for the 1D an- veloped both in theory [281, 282], and in experi- tiferromagnetic spin chains with integer or half- ments [283]. The recent wave of very successful ex- integer spins, respectively. Ultracold spinor gases periments creating artificial or synthetic magnetic [254, 255] might help to study these questions. fields employing laser induced gauge fields [81, 82] Again, particularly interesting are in this context that are achieved by using spatial dependent op- spinor gases in optical lattices [256–259], where in tical coupling between different internal states of the strongly interacting limit the Hamiltonian re- atoms. Such approach is free from rotational re- duces to a generalized Heisenberg Hamiltonian. strictions. Very recently, this successful wave have Spinor condensates in optical lattices have been invested also lattice experiments. Paradigmatic 2D addressed experimentally for ferromagnetic [260– models displaying a topological insulating behav- 263] and antiferromagnetic [264] interactions. Us- ior like the Hofstadter [284] and the Haldane [285] ing Feshbach resonances [234] and varying the lat- models have been realized experimentally in shal- tice geometry one should be able in such systems to low harmonic traps in [286, 287] and[288], respec- generate a variety of regimes and quantum phases, tively, by exploiting superlattices and Bragg pulses. including the most interesting antiferromagnetic The Hofstadter model has been also realized in lad- (AF) regime. García-Ripoll et al.[165] proposed to ders and slabs, that to say in lattices with sharp use a duality between the antiferromagnetic (AF) boundaries in one narrow dimension. Such di- = − and ferromagnetic (F) Hamiltonians, HAF HF , mension can be real as in the experiment by [289] which implies that minimal energy states of HAF or synthetic [290], that to say with the rungs of the are maximal energy states of HF , and vice versa. ladder formed by the internal spin states of atoms, Since dissipation and decoherence are practically which provide an extra dimension to the system negligible in such systems, and affect equally both [291]. The synthetic Hofstadter slabs has been real- ends of the spectrum, one can study AF physics ized experimentally for bosons [292] and fermions with HF , preparing adiabatically AF states of in- [293] and the edge currents observed for the first terest. Ytterbium fermionic atoms with N differ- time via spin-dependent measurements. Real and ent spin states in an optical lattices implementing synthetic Hofstadter ladders and slabs offer a very the fermionic SU(N) Hubbard model have been re- promising route to the observation of many-body cently experimentally investigated [265]. quantum Hall physics, both for bosons [294, 295] and fermions [296, 297]. • Fractional quantum Hall states. Since the famous work of Laughlin [266], there has been enormous • Lattice gauge fields. Gauge theories, and in par- progress in our understanding of the fractional ticular lattice gauge theories (LGT) [298] are fun- quantum Hall effect (FQHE) [267]. Nevertheless, damental for both high energy physics and con- many challenges remain open like the direct ob- densed matter physics, and despite the progress servation of the anyonic character of excitations or of our understanding of LGT, many questions in the observation of other kinds of strongly corre- this area remain open (see e.g. [299]). Physics of lated states. FQHE states might be studied with cold atoms might help here in two aspects: “ar- trapped ultracold rotating gases [268, 269]. Rota- tificial” non Abelian magnetic fields may be cre- tion induces there effects equivalent to an “artifi- ated in lattice gases via appropriate control of the cial” constant magnetic field directed along the ro- hopping matrix elements [300] or in trapped gases tation axis. There are proposals about how to de- using effects of electromagnetically induced trans- tect directly fractional excitations in such systems parency [301]. One of the most challenging tasks [270]. Optical lattices might help in this task in in this context concerns the possibility of realiz- two aspects. First, FQHE states of small systems ing generalizations of Laughlin states with pos- of atoms could be observed in a lattice with rotat- sibly non Abelian fractional excitations. Another ing site potentials, or an array of rotating micro- challenge concerns the possibility of “mimicking” traps —cf. [271],[272],[273] and [274] and refer- the dynamics of gauge fields. In fact, dynamical ences therein—. Second, ”artificial” magnetic field realizations of U(1) Abelian gauge theory, that in- 9

volve ring exchange interaction in a square lattice nant dipole interactions. More recently, also dipo- [302], or 3 particle interactions in a triangular lat- lar gases of Dysprosium [333] and Erbium [334] tice [303, 304] have been also proposed. In the last have been cooled up to reach quantum degener- four years the effort of simulating LGT in optical acy. Very recently, dipolar gases [335] and polar lattices has received new impulse. Most of the at- molecules [336] have been used to simulate quan- tention has focused on gauge magnets or link mod- tum magnetism. Dipolar BECs and BCS states of els [305–307], which can be viewed as spin version trapped gases are expected to exhibit very inter- of ordinary Hamiltonian LGT and, as such, are eas- esting dependence on the trap geometry. Dipo- ier to be simulated. Both Abelian and non-Abelian lar ultracold gases in optical lattices, described by LGT can be simulated by exploiting angular mo- extended Hubbard models, should allow to re- mentum conservation [308, 309],SU(N)-invariant alize various quantum insulating “solid” phases, interaction in earth-alkali like atoms [310, 311], such as phase checkerboard (CB), and superfluid or the long-range interaction induced by Rydberg phases such as supersolid (SS) phase [337–339]. atoms [312, 313]. The proposed simulators would Particularly interesting in this context are the ro- be capable to probe the confinement of charges, for tating dipolar gases (RDG) . Bose-Einstein conden- instance, by measuring the string tension. Similar sates of RDGs exhibit novel forms of vortex lat- simulators have been proposed also for supercon- tices: square, “stripe crystal”, and “bubble crys- ducting qubit [314] and trapped ions [315], where tal” lattices [340].[341] have demonstrated that the string breaking in Schwinger model has been very pseudo-hole gap survives the large N limit for the recently experimentally demonstrated with four Fermi RDGs, making them perfect candidates to ions [77]. These developments in quantum simu- achieve the strongly correlated regime, and to real- lation has triggered also parallel developments in ize Laughlin liquid at filling ν = 1/3, and quantum classical simulation both in 1D [316–318]and2Dor Wigner crystal at ν ≤ 1/7 [342] with mesoscopic more [319, 320]. number of atoms N ≃ 50 − 100.

• Superchemistry. This is a challenge of quantum • Wigner crystals or self-assembled lattices. Wigner, chemistry, rather than condensed matter physics: or Coulomb type of crystals are predicted to be to perform a chemical reaction in a controlled way, formed due to long range repulsive dipolar atom- by using photoassociation or Feshbach resonances atom or molecule-molecule interactions [343–345], from a desired initial state to a desired final quan- or ion-ion Coulomb interactions in the absence of tum state. Jaksch et al.[321] proposed to use a Mott a lattice [64]. insulator (MI) with two identical atoms, to cre- ate via photoassociation, first a MI of homonuclear Many of the above mentioned challenges are dis- molecules , and then a molecular SF via “quantum cussed discussed in the subsequent chapters of our melting” . A similar idea was applied to heteronu- book [1]. The interested reader should, however, ex- clear molecules [322] in order to achieve molecular tend her/his knowledge by turning toward the excel- SF. Bloch’s group have indeed observed photoasso- lent books of Pitaevskii and Stringari [9] and Pethick ciation of 87Rb molecules in a MI with two atoms and Smith [346] on Bose-Einstein Condensation in the per site [323], while Rempe’s group has realized weakly interacting regime, or to the books of Fetter and the first molecular MI using Feshbach resonances Walecka [347] or X.G. Wen [348] for many body and [54]. Formation of three-body Efimov trimer states quantum field theory, among others. was observed in trapped Cs atoms by Grimm’s group [324]. This process could be even more ef- IV. CONCLUSIONS ficient in optical lattices [325]. An overview of the subject of cold chemistry can be found in [326], and in particular about cold Feshbach molecules EU as well many other countries have concentrated in [327]. Control and creation of deeply bound recently considerable efforts to support quantum infor- molecules in the presence of an optical lattice has mation science and technology. EU in particular will been reported in [328]. launch a Quantum Flagship, which will seek for techno- logical advances in four pillars of the QI science; Quan- • Ultracold dipolar gases. Some of the most fasci- tum Computing and Simulations, Quantum Communi- nating experimental and theoretical challenges of cations, Quantum Sensing and Metrology, and Quantum the modern atomic and molecular physics con- information Theory and Software. cern ultracold dipolar quantum gases —for re- While universal quantum computers remains still views, see [329],[330] and [331]—. The recent ex- very challenging, quantum simulators already exist in perimental realization of the dipolar Bose gas of the labs. Many of those are analogue or digital simula- Chromium [332], and the progress in trapping and tors of “interesting” quantum phenomena. In the recent cooling of dipolar molecules [253] have opened the year, however, a lot of effort has been devoted to quan- path towards ultracold quantum gases with domi- tum annealers, like D- wave machines that are designed 10 to solve classical NP-complete or at least ultra-complex as predicted by Einstein about a century ago [358, 359], problems [202, 349, 350]. Quantum annealers are per- the impressive progresses in the optical clock standards haps the first quantum computing devices that have the [360] are expected to lead in the next decade to direct chance to enter real technology and change our everyday tests of fundamental laws of physics by proving tighter life. In fact, a decisive progress toward definitive success and tighter bounds on the stability of fundamental con- of quantum computing devices may be achieved by inte- stants [361]. grating the different platforms [351] that are best suited Thus, exciting time and challenges are expecting the for different tasks. The development of such integrated quantum optics community in the following years that structure would incarnate fully the spirit of the Quan- require a common effort. tum Flagship, and has quantum optics in its heart. Ones of the most intriguing applications of quantum simula- Quantum workers of the world, unite! tors 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 This is not the final struggle, but... Gauge Theories is especially fascinating and may lead to Let us group together, and tomorrow breaking the 20th century distinction between low and The Quantum Internationale high energy physics. Such simulators may serve also Will be the human race. for testing holographic principle [353]. In fact, quantum simulation of high energy phenomena and gravitation We acknowledge financial support from John Temple- [354–356] is not the only way in which quantum optics ton Foundation, European Union (EU IP SIQS, EU Grant and quantum technologies may have a revolutionary im- PROACT QUIC, EU STREP EQUAM), the European Re- pact over our understanding of Nature at all scales. In- search Council (ERC AdG OSYRIS), and the Spanish deed, in addition to the celebrate success of the inter- MINECO (National Plan projects FOQUS – FIS2013- ferometer LIGO in detecting gravitational waves [357] 46768 and Q-TRIP – FIS2014-57460-P) and the General- itat de Catalunya AGAUR (2014 SGR 874 and SGR2014- 1639).

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