Parametric amplification of vacuum fluctuations is crucial in modern quantum optics, enabling the creation of squeezing and entanglement. We demonstrate the parametric amplification of vacuum fluctuations for matter waves using a spinor F = 2 87 Rb condensate. Interatomic interactions lead to correlated pair creation in the mF = ±1 states from an initial unstable mF = 0 condensate, which acts as a vacuum for mF = 0. Although this pair creation from a pure mF = 0 condensate is ideally triggered by vacuum fluctuations, unavoidable spurious initial mF = ±1 atoms induce a classical seed which may become the dominant triggering mechanism. We show that pair creation is insensitive to a classical seed for sufficiently large magnetic fields, demonstrating the dominant role of vacuum fluctuations. The presented system thus provides a direct path towards the generation of non-classical states of matter on the basis of spinor condensates.Parametric amplification of vacuum fluctuations in nonlinear media [1] plays a crucial role in the investigation and application of non-classical states of light. These states have revolutionized the field of quantum optics in the past decades. Since the first observation of squeezed light [2], these non-classical states of light have become a valuable tool in modern optics, e.g. for the enhancement of modern interferometers [3]. Similarly, the production of entangled photon pairs [4] has triggered an ongoing series of fundamental tests of modern quantum mechanics [5,6] and has many possible applications for quantum computing [7]. The tools developed for the production and manipulation of ultracold neutral atoms now bring many of these seminal investigations within the scope of experiments with matter waves. In this sense, the production of number-squeezed Bose-Einstein condensates (BECs) [8] and spin squeezed thermal clouds [9] has been recently demonstrated.Spinor BECs, consisting of atoms with non-zero spin F , provide an optimal non-linear medium for the production of non-classical states of matter. In these systems, inter-particle interactions lead to a coherent population transfer between different Zeeman m F sublevels (spin dynamics). The case of a sample prepared in m F = 0 (represented by |0 ) is particularly interesting. In that case the initial stages of the spin dynamics are characterized by the production of correlated atom pairs in |±1 [10,11], resembling the production of EinsteinPodolsky-Rosen (EPR) pairs in optical parametric down conversion [5].Ideally a pure initial |0 BEC acts as a vacuum for atoms in m F = 0. Hence pair creation into |±1 can be understood as a parametric amplification of quantum vacuum fluctuations. However, parametric amplifiers are exponentially sensitive to any spurious initial seed in the amplified modes, which may easily dominate the effect of vacuum fluctuations. Despite careful purification proce-0

Parametric amplification of quantum fluctuations constitutes a fundamental mechanism for spontaneous symmetry breaking. In our experiments, a spinor condensate acts as a parametric amplifier of spin modes, resulting in a twofold spontaneous breaking of spatial and spin symmetry in the amplified clouds. Our experiments permit a precise analysis of the amplification in specific spatial Bessel-like modes, allowing for the detailed understanding of the double symmetry breaking. On resonances that create vortex-antivortex superpositions, we show that the cylindrical spatial symmetry is spontaneously broken, but phase squeezing prevents spin-symmetry breaking. If, however, nondegenerate spin modes contribute to the amplification, quantum interferences lead to spin-dependent density profiles and hence spontaneously formed patterns in the longitudinal magnetization.

We analyze the spinor dynamics of a 87 Rb F = 2 condensate initially prepared in the mF = 0 Zeeman sublevel. We show that this dynamics, characterized by the creation of correlated atomic pairs in mF = ±1, presents an intriguing multi-resonant magnetic field dependence induced by the trap inhomogeneity. This dependence is directly linked to the most unstable Bogoliubov spin excitations of the initial mF = 0 condensate, showing that, in general, even a qualitative understanding of the pair creation efficiency in a spinor condensate requires a careful consideration of the confinement.Spinor Bose-Einstein condensates (BECs), consisting of atoms with non-zero spin, constitute an ideal scenario to investigate the interplay between internal and external degrees of freedom in a multi-component superfluid. The competition between spin-dependent collisional interactions, Zeeman effect, and inhomogeneous trapping results in an exciting range of fundamental phenomena. As a result, spinor BECs have attracted large attention since the pioneering experiments in optical traps [1] concerning their ground state properties [2,3,4,5] and the spinor dynamics induced by the spin-changing collisions, which allow for a coherent transfer between different spin components [6,7].Spinor BECs also provide exciting perspectives as novel sources of non-classical states of matter. In this sense, condensates initially prepared in the m F = 0 Zeeman sublevel are especially fascinating [6,7,8,9]. In that case, the creation of correlated pairs results in the growth of macroscopic populations in m F = ±1. This amplification process is ideally triggered by quantum spin fluctuations [10]. Interestingly, it closely resembles parametric amplification in optical parametric down conversion [11], opening exciting new routes for matter-wave squeezing and atomic Einstein-Podolsky-Rosen entanglement experiments [12,13].Correlated pair creation, and in general any spinor dynamics, demands a significant rate of spin-changing collisions. In typical experiments these collisions are suppressed by the quadratic Zeeman effect (QZE) already in the presence of moderate magnetic fields [6]. However, the influence of the QZE at low fields is far from trivial [8,14,15,16,17]. In particular, spin-mixing can reach a pronounced maximum for low but finite fields. This resonance, contrary to those discussed below, has a non-linear character and has been explained in terms of phase matching [16].The understanding of the magnetic-field dependence of the pair creation efficiency is hence crucial for the characterization of novel spinor-based sources of non-classical matter waves. In this Letter we show that this dependence generally presents an intriguing non-monotonous character which is crucially determined by the trap inhomogeneity and cannot be explained from the physics of homogeneous BECs [18]. The pair creation efficiency is directly linked to the instability rate, which characterizes the exponential growth of the most unstable spin excitations of the initial BEC in m F...

We present an analytical model for the theoretical analysis of spin dynamics and spontaneous symmetry breaking in a spinor Bose-Einstein condensate (BEC). This allows for an excellent intuitive understanding of the processes and provides good quantitative agreement with the experimental results of Scherer et al. [Phys. Rev. Lett. 105, 135302 (2010)]. It is shown that the dynamics of a spinor BEC initially prepared in an unstable Zeeman state m F = 0 (|0 ) can be understood by approximating the effective trapping potential for the state |±1 with a cylindrical box potential. The resonances in the creation efficiency of these atom pairs can be traced back to excitation modes of this confinement. The understanding of these excitation modes allows for a detailed characterization of the symmetry-breaking mechanism, showing how a twofold spontaneous breaking of spatial and spin symmetry can occur. In addition, a detailed account of the experimental methods for the preparation and analysis of spinor quantum gases is given.

Spin-changing collisions may lead under proper conditions to the parametric amplification of matter waves in spinor Bose-Einstein condensates. Magnetic dipole-dipole interactions, although typically very weak in alkalimetal atoms, are shown to play a very relevant role in the amplification process. We show that these interactions may lead to a strong dependence of the amplification dynamics on the angle between the trap axis and the magnetic-field orientation. We analyze as well the important role played by magnetic-field gradients, which also modify strongly the amplification process. Magnetic-field gradients, hence, must be carefully controlled in future experiments, in order to observe clearly the effects of the dipolar interactions in the amplification dynamics.