Quantum simulation is certainly one of the most promising quantum technology, but it is a vast domain, in constant evolution, and where dramatic innovations appear all the time. So when you watch that section, unexpected progress may have happened. Although this is a quantum optics course, and there are significant results with photons or superconducting devices, I will rather take here the example of ultracold atoms simulators, which seem to be at a more advanced stage, comparable to the stage of ultracold ions. In 1982, Feynman realized the extraordinary character of entanglement. Among the dramatic departures from classical ideas, he insisted that when the number of entangled quantum object increases, the size of their Hilbert space increases exponentially with the number of entangled objects. Consider the case of a quantum bit: the simplest possible quantum object, whose state belongs to a two-dimensional space spanned by zero and one. In the case of photons, it corresponds to two orthogonal polarizations, for instance x and y. The states of two qubits belong to a space of dimension four spanned by 00, 01, 10 and 11. The generalization is clear: for n entangled qubits, the space of states has dimension two to the power n, which means that 10 entangled qubits belong to a space of dimension 1,024, about 1,000. For 20 qubits, the dimension is one million, and it is one billion for 30 entangled qubits. I can stop here. You understand that if we want to describe the electrons of a complicated organic molecule, for instance a drug, or the electrons of a small piece of matter, it would demand more data than any computer can handle. In order to describe such a system and study some of its properties, Feynman then suggested that the only possibility is to use a quantum system with many entangled quantum objects, whose space of state will be as large as a space of state of the system we want to study. The simplest realization of that idea is a system made of quantum objects that are placed in a situation analog to the system we want to study. An interesting example is the case of electrons in a crystal, that is to say, in a periodic potential, replaced by atoms in a potential created by laser beams forming a standing wave. It is possible to simulate situations in one, two, or three dimensions. Laser cooling techniques allow one to have a thermal energy of the atoms small compared to the amplitude of the potential, and quantum statistics play a major role. So the nature of the atoms, fermions or bosons, is essential. In fact, one can choose between fermions or bosons by selecting the right isotope of atoms. Lithium or potassium are much used atoms for that purpose. For instance, lithium whose number of electrons is three, has two isotopes. Lithium seven is a boson, since the total number of nucleons plus electrons is even. While lithium six is a fermion, since three plus six is odd. In the case of bosonic atoms, it is possible to cool them at a low enough temperature to obtain a Bose-Einstein condensate, with negligible thermal excitation. The many-body system of interacting atoms can be put in its fundamental state. It is not the case with fermions, where the best cooling technique lead to a non negligible fraction of the Fermi temperature, and the thermal component plays a significant role. The interactions between the quantum particles can be simulated by interactions between the atoms which can be tuned at will, in sign and strength, with the magnetic field. This is the celebrated Feschbach resonance. At the end of the day, it is possible to mimic several paradigmatic condensed matter many-body problems too complicated to be solved on a classical computer. You may ask the question: what is the benefit of experimenting on an analog situation compared to a direct experimental study of the problem we are interested in? There are many good reasons of which I will cite a few only, comparing electrons in a solid to a simulator with ultracold atoms. Firstly, it is easy to observe atoms, thanks to fluorescence imaging, while it is very difficult to directly observe electrons in a piece of matter. Also, in a simulator, one can easily control and change the parameters of the problem investigated. For instance, the amplitude of the potential or the density of the quantum particles. This is rarely the case in the original condensed matter problem. Last and not least, with atoms trapped with lasers, it is possible to instantly switch off the potential, giving access to the distribution of velocities by observing the distribution of the atoms after a long enough time of flight. A celebrated example of a many-body problem investigated with a cold atom simulator, is the Mott quantum phase transition. In the 1960s, Nevill Mott, who was later knighted and became Sir Nevill, invented a model to render an account of the fact that some conductors experience a phase transition from conductor to insulator, when the level of interactions increases. His model starts from the tight binding model of the electrons in the potential wells of the periodic potential, in which one has a conductor because of tunneling between sites. He then adds a repulsive interaction between the electrons, modeled by an interaction energy when two electrons are on the same site. When one changes the density of the electrons, one has a phase transition between the conducting and the insulating situation, due to the competition between the tunneling, which favors conduction, and the repulsive interaction, which hampers conduction. That phase transition is not driven by thermal excitations but by quantum fluctuations associated with the localization of the particles in the potential wells. The Mott transition is thus a quantum phase transition. Electrons are fermions, but the Mott transition model has also been invoked for the superfluid to normal fluid transition in liquid helium which is made of bosons. The Bose-Hubbard Hamiltonian renders an account of the competing energy terms, and takes into account the bosonic quantum statistics. For all these models, the theoretical methods appropriate for weak interacting systems, fail to describe the situation. In general, there is no exact theoretical solution of the problem. Numerical methods give interesting results, but they rely on approximations, and it is thus highly desirable to test the model itself. This has been done in 2002 in Munich, by Immanuel Bloch and his colleagues, where a convincing ultracold atoms simulation has allowed the authors to observe and study quantitatively the main characteristics of the transition. The figure here illustrates the possibilities of cold atoms simulations. Changing the depth of the periodic lattice, simply by changing the intensity of the laser standing wave, one crosses a Mott transition. This is shown by the observation of the velocity distribution of the atoms, thanks to a long enough time of flight, following the switching off of the lattice. In the superconducting phase, the atomic wave functions are extended over many sites, and are periodic. Their Fourier transform, whose squared modulus is the velocity distribution, as those sharp peaks. For a larger depth of potential wells, the atoms are in the Mott insulator phase, with one atom at each site. Then the various one atom wave functions are not coherent, and their bell-shaped Fourier transforms add incoherently. The whole process of the Mott transition, is beautifully shown directly in that experiment, which has convinced a broad community of physicists of the interests of cold atom quantum simulators. I cannot resist citing another example of a cold atoms simulation, showing directly a quantum transition from a conducting state to an insulator state. It is called the Anderson Localization, and the model was developed at about the same time as the Mott model, in the end of the 1950's. Phil Anderson and Nevil Mott received the Nobel Prize the same year in 1978. In contrast to the Mott model, which describes quantum particles in a perfect crystal, the Anderson model describes the behavior of quantum particles in a disordered potential. A classical particle would diffuse at large distance provided its energy is larger than the maxima of the potential. But for a quantum particle, when the disorder is large enough although small compared to the energy of the particle, the Anderson model predicts that the propagation is totally stopped due to a subtle destructive interference in the multiply-scattered matter-wave. The transition from conductor to insulator when the disorder increases is here also a quantum phase transition. There is no exact quantitative theory allowing one to calculate several characteristics of the transition, for instance, the value of the energy of the transition or its critical exponents. Since only approximated theories exist, it is important to test them. Indirect evidences of that effect had been accumulated over decades in condensed matter physics, but in 2007, an ultra cold atoms quantum simulator allowed us to directly image the localized wave function, predicted by the theory in the localized phase, and to observe the transition to extended wave functions, when the amplitude of the disorder decreases. The disordered potential was created with a laser beam passing through a scattering plate, to produce a so-called laser speckle pattern, a well-known random distribution of light intensity. Several properties of Anderson localization are still under study with our cold atom simulator which can operate in one, two, and three dimensions. A new type of cold atoms simulator recently demonstrated is very promising not only to study physics problems that it emulates, but also as a programmable quantum computer as you will learn in the next section. This simulator is based on the possibility to load and manipulate single atoms in an optical tweezer, that is to say, a strongly focused laser beam able to attract particles at a focus. Art Ashkin received the 2018 Nobel Prize for the invention of the optical tweezer, 30 years earlier and its applications to biological objects. Fifteen years later, my colleague Philippe Grangier and his collaborators, found that with a strongly focused laser beam, one and only one atom can be trapped in a tweezer. As you learned in Quantum Optics 1, having a single quantum object opens many possibilities compared to a statistical distribution. Here also, there is a big benefit when you are sure that you have single quantum objects. Spatial light modulators, the so-called SLM, offer the possibility to create many tweezers with one atom in each and to move individually each atom. As shown by Antoine Browaeys and his collaborators, one can then create arrays of a few tens of atoms with any shape, in one, two or three dimensions. Really any shape. Combined with the possibility to tailor entanglement between these atoms by exciting them to strongly interacting states called Rydberg states, one can simulate situations where dozens of quantum particles interact in a sophisticated way. Difficult problems of condensed matter can be addressed with these simulators, such as the celebrated Ising model of interacting spin one-half particles disposed on a periodic lattice and placed in a magnetic field. This is an example of atoms in a triangular or a square lattice in two dimensions. In this simulator, spin 1/2 particles in a magnetic field are simulated by atoms evolving between two levels only, the ground state and one Rydberg state. Some Ising models seem to describe quite well the properties of some types of magnetism such as ferromagnetism or anti-ferromagnetism. But except for specific cases, there is no exact solution to the problem of finding the steady-state of the system, that is to say, the lowest eigenstate of the Hamiltonian describing it. Moreover, perturbative approaches valid for weak interactions fail, because they cannot describe a situation with strong correlations between the neighboring particles. In a ferromagnetic material, for instance, first neighbors are parallel, while in anti-ferromagnetism first neighbors are anti-parallel. Numerical solutions of the problem demand a time increasing exponentially with the size of the sample and would take too long time for more than a few tens of particles. In contrast, simulators with up to 50 atoms have already produced interesting results, as shown by this simulation of an anti-ferromagnetic situation. There does not seem to be any major problem to extend it to hundreds of atoms. Several groups in the world work on such systems and one should enter soon in the domain where the solutions found by the simulator are impossible to compare to the solutions found by traditional methods. It is thus important to compare the results found with various kind of simulators, not only cold atoms, but also cold ions or other kinds of platforms.