To gain deeper insight into the dynamics of complex quantum systems we need a quantum leap in computer simulations. We can not translate quantum behaviour arising with superposition states or entanglement efficiently into the classical language of conventional computers. The final solution to this problem is a universal quantum computer [1], suggested in 1982 and envisioned to become functional within the next decade(s); a shortcut was proposed via simulating the quantum behaviour of interest in a different quantum system, where all parameters and interactions can be controlled and the outcome detected sufficiently well.Here we study the feasibility of a quantum simulator based on trapped ions [2]. We experimentally simulate the adiabatic evolution of the smallest non-trivial spin system from the paramagnetic into the (anti-)ferromagnetic order with a quantum magnetisation for two spins of 98%, controlling and manipulating all relevant parameters of the Hamiltonian independently via electromagnetic fields. We prove that the observed transition is not driven by thermal fluctuations, but of quantum mechanical origin, the source of quantum fluctuations in quantum phase transitions [3]. We observe a final superposition state of the two degenerate spin configurations for the ferromagnetic (|↑↑ +|↓↓ ) and the anti-ferromagnetic (|↑↓ + |↓↑ ) order, respectively. These correspond to deterministically entangled states achieved with a fidelity up to 88%.Our work demonstrates a building block for simulating quantum spin-Hamiltonians with trapped ions. The method has potential for scaling to a higher number of coupled spins [2].

We study phonons in a dynamical chain of ions confined by a trap with a time-dependent (axial) potential strength and demonstrate that they behave in the same way as quantum fields in an expanding or contracting Universe. Based on this analogy, we present a scheme for the detection of the analogue of cosmological particle creation which should be feasible with present day technology. In order to test the quantum nature of the particle creation mechanism and to distinguish it from classical effects such as heating, we propose to measure the two-phonon amplitude via the 2nd red sideband transition and to compare it with the one-phonon amplitude (1st red sideband).

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