A mechanism leading to spontaneous breaking of the space symmetry, and to the appearance of spatial patterns in the most recently created state of matter, the Bose-Einstein condensate (BEC), is proposed. The pattern formation are due to a periodic modulation in time of the scattering length of the condensed atoms. Experimentally this can be achieved via the Feshbach resonance, or, for low dimensional (1D or 2D) condensates, also by modulation of the trapping potential along a spatial coordinate normal to the "working" space. The parametric excitation of patterns in condensates by considering the mean-field Gross-Pitaevskii equation, and its dissipative generalization are investigated.
We show that metastable phases of an antiferromagnetic spin-1 condensate in a simple model with pure contact interactions can exhibit a rotonlike minimum in the excitation spectrum. The introduction of a magnetic field gives rise to the instability of roton modes, which can lead to spontaneous emergence of regular periodic, polygonal, polyhedral, or crystalline patterns, as shown in numerical simulations within the truncated Wigner approximation. An explanation of the occurrence of rotonlike instability is given based on the energy and spin conservation laws.
We study the formation of transient Faraday patterns and spin textures in driven quasi-one-dimensional and quasi-two-dimensional spin-1 Bose-Einstein condensates under the periodic modulation of
s-wave scattering lengths
a0 and
a2, starting from the anti-ferromagnetic phase. This phase is characterized by a Bogoliubov spectrum consisting of three modes: one mode is gapped, while the other two are gapless. When
a0 is modulated and half of the modulation frequency lies below the gapped mode, density and spin Faraday patterns emerge. In that case, in quasi-one-dimension, the spin texture is characterized by periodic domains of opposite
z-polarizations. When driven above the gap, the spin texture is characterized by random orientations of spin vectors along the condensate axis. Qualitatively new features appear in the driven quasi-two-dimensional condensate. For instance, when driven above the gap, the spin textures are characterized by anomalous vortices and antivortices that do not exhibit phase winding in individual magnetic components. Below the gap, the spin texture exhibits irregular ferromagnetic patches with opposite polarizations. The spatial spin-spin correlations in quasi-one-dimension exhibit a Gaussian envelope, whereas they possess a Bessel function dependence in quasi-two-dimension. Under the
a2-modulation, the density patterns dominate irrespective of the driving frequency, unless the spin-dependent interaction strength is sufficiently smaller than that of the spin-independent interaction. The intriguing scenario of competing instability can emerge when both scattering lengths are simultaneously modulated. Finally, we show that the competing instabilities result in a complex relationship between the population transfer and the strength of the quadratic Zeeman field, while keeping all other parameters constant.
With a mean-field treatment, the spin tunneling of spin-1 Bose-Einstein condensates in a linearly sweeping magnetic field is investigated. We focus on a typical alkali Bose atoms 87Rb condensate and study its tunneling dynamics under different sweeping rates of external magnetic field. In the adiabatic (i.e., slowly sweeping) and sudden (i.e., fast sweeping) limits, no tunneling is observed. In the case of moderate sweeping rates, the tunneling dynamics is found to be very sensitive to the sweeping rate, showing a chaotic-like tunneling regime. By magnifying the apparently chaotic regime, however, we find interestingly that the plot becomes resolvable at a resolution of 10-8T/s where the tunneling probability with respect to the sweeping rate shows a regular pseudo-periodic pattern. Moreover, a conserved quantity standing for the magnetization in experiments is found which can dramatically affect the above picture of the spin tunneling. Theoretically, we give a reasonable interpretation to the above findings.
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
mF=0 ($\left|0\right\ensuremath{\rangle}$) can be understood by approximating the effective trapping potential for the state $\left|\ifmmode\pm\else\textpm\fi{}1\right\ensuremath{\rangle}$ 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.
The onset of pattern formation in a spatially homogeneous system subjected to external driving is an important topic in various scientific fields. A celebrated classical example is the Faraday instability, where a vertically oscillated fluid surface undergoes a parametric resonance, giving rise to standing waves that self-organize into regular spatial patterns. Bose-Einstein condensates (BECs) provide an ideal quantum-mechanical platform for studying pattern-forming mechanisms due to their exceptional degree of experimental control. As a compressible state of quantum matter, a condensate responds sensitively to external perturbations, including time-periodic modulation of interactions, trapping potentials, or external fields. These features make BECs particularly well suited for exploring driven nonequilibrium phases and pattern formation. In this chapter, we review the remarkable progress achieved in this field over the past two decades. We begin with the first theoretical proposal predicting parametric instabilities and emergent Faraday waves in driven condensates. We then discuss key experimental and theoretical breakthroughs that confirmed these predictions and refined the understanding of the underlying mechanisms. This line of research has culminated in the recent observation of a stabilized square lattice pattern in a periodically driven BEC confined in a two-dimensional geometry. This driven superfluid state with superposed density modulation was shown to exhibit some features of a supersolid state.