Large spin systems can exhibit unconventional types of magnetic ordering, different from the ferromagnetic or N\'eel-like antiferromagnetic order commonly found in spin 1/2 systems. Spin-nematic phases, for instance, do not break time-reversal invariance, and their magnetic order parameter is characterized by a second-rank tensor with the symmetry of an ellipsoid. Here we show direct experimental evidence of spin-nematic ordering in a spin 1 Bose-Einstein condensate of sodium atoms with antiferromagnetic interactions. In a mean field description this order is enforced by locking the relative phase between spin components. We reveal this mechanism by studying the spin noise after a spin rotation, which is shown to contain information hidden when looking only at averages. The method should be applicable to high-spin systems in order to reveal complex magnetic phases.
Large spin systems can exhibit unconventional types of magnetic ordering, different from the ferromagnetic or N'eel-like antiferromagnetic order commonly found in spin 1/2 systems. Spin-nematic phases, for instance, do not break time-reversal invariance, and their magnetic order parameter is characterized by a second-rank tensor with the symmetry of an ellipsoid. Here we show direct experimental evidence of spin-nematic ordering in a spin 1 Bose-Einstein condensate of sodium atoms with antiferromagnetic interactions. In a mean field description this order is enforced by locking the relative phase between spin components. We reveal this mechanism by studying the spin noise after a spin rotation, which is shown to contain information hidden when looking only at averages. The method should be applicable to high-spin systems in order to reveal complex magnetic phases.
Exotic types of magnetic order and phases resulting from collective behaviour of quantum spins are an important focus of many-body physics. Nematic or quadrupolar ordering of spins is one such example [1], which breaks O(3) rotational symmetry, has no magnetic moment and is analogous to the well-known ordering of molecules in nematic phases of liquid crystals [2]. Spin nematic phases have been posited for a variety of condensed matter systems including frustrated quantum magnets [3–5], and heavyfermion [6] and iron-based superconductors [7], although they are challenging to detect directly [8]. Spin-1 atomic Bose-Einstein condensates [9– 11] provide a natural system to investigate spinnematic quantum phases with a key advantage that the nematic tensor is directly measurable. Here, we measure spin-nematic fluctuations in a spin-1 condensate following a quench through a nematic to ferromagnetic quantum phase transition and observe quadrature squeezing in the variance of the fluctuations up to −8.3 −0.7 dB (−10.3 −0.9 dB corrected for detection noise) below the standard quantum limit. Previous work in atomic squeezing has employed two-level systems exhibiting SU(2) symmetry on the Bloch sphere [12, 13], while the squeezing observed here is an example of squeezing in an SU(3) system. These results demonstrate spin-nematic dynamics in the quantum regime and could form the basis of a quantum-enhanced magnetometer.
In this thesis, we investigate strongly correlated phenomena in quantum spin systems. In the first part of this work, we study geometrically frustrated antiferromagnets (AFMs). Generalizing the SU(2) Heisenberg Hamiltonian to Sp(N) symmetry, we obtain, in the large-N limit, the mean-field phase diagrams for the planar pyrochlore and cubic AFMs. We then use gauge theories to consider fluctuation effects about their respective mean-field configurations. We find, in addition to conventional Neel states, a plethora of novel magnetically disordered phases: two kinds of spin liquids, Z2 in 2+1D and U(1)in 3+1D, and several valence bond solids such as two and three-dimensional plaquette and columnar singlet states. We use the same approach to study the diamond lattice AFM which possesses extended classical ground state degeneracy. We demonstrate that quantum and entropic fluctuations lift this degeneracy in different ways. In the second part of the thesis, we study ultracold spinor atoms confined in optical lattices. We first demonstrate the feasibility of experimental realization of rotor models using ultracold spin-one Bose atoms in a spin-dependent and disordered optical lattice. We show that the ground state of such disordered rotor models with quadrupolar interactions can exhibit biaxial nematic ordering in the disorder-averaged sense, and suggest an imaging experiment to detect the biaxial nematicity in such systems. Finally, using variational wavefunction methods, we study the Mott phases and superfluid-insulator transition of spin-three bosons in an optical lattice with an anisotropic two dimensional optical trap. We chart out the phase diagrams for Mott states with n = 1 and n = 2 atoms per lattice site. We show that the long-range dipolar interaction stabilizes a state characterized by antiferromagnetic chains made of ferromagnetically aligned spins. We also obtain the mean-field phase boundary for the superfluid-insulator transition, and show that inside the superfluid phase and near the superfluid-insulator phase boundary, the system undergoes a first order antiferromagnetic-ferromagnetic spin ordering transition.
We have experimentally observed the emergence of spontaneous antiferromagnetic spatial order in a sodium spinor Bose-Einstein condensate that was quenched through a magnetic phase transition. For negative values of the quadratic Zeeman shift, a gas initially prepared in the F=1, m(F)=0 state collapsed into a dynamically evolving superposition of all three spin projections, m(F)=0, ±1. The quench gave rise to rich, nonequilibrium behavior where both nematic and magnetic spin waves were generated. We characterized the spatiotemporal evolution through two particle correlations between atoms in each pair of spin states. These revealed dramatic differences between the dynamics of the spin correlations and those of the spin populations.
Spinor Bose-Einstein condensates under external magnetic fields exhibit well-characterized spin domains of its ground state due to spin-dependent interactions. At low temperatures, collision-induced spin-mixing instabilities may promote the condensate to dwell into metastable states occurring near the phase boundaries. In this paper, we study theoretically the metastable spin-phase diagram of a spin-1 antiferromagnetic Bose-Einstein condensate at zero and finite temperatures. The approach makes use of Hartree-Fock theory and exploits the symmetry of the Hamiltonian and of the order parameters yielding a closed system of transcendental equations for the free energy, fully avoiding the use of self-consistency methods. Our results are consistent with recent experiments and allow us to explain qualitatively the different types of observed quench dynamics. In addition, we found that similar phenomena should occur in antiferromagnetic spinor condensates with a sudden change in the temperature. It is shown also that the increase in temperature induces a traceable shift of the ferromagnetic-polar transition boundary, behavior previously not noted by self-consistent mean-field calculations.
We develop a symmetry classification scheme to find ground states of pseudo-spin-21, spin-1, and spin-2 spin-orbit-coupled spinor Bose-Einstein condensates and show that as the SO(2) symmetry of simultaneous spin and space rotations is broken into discrete cyclic groups, various types of lattice structures emerge in the absence of a lattice potential. Examples include two different kagome lattices for pseudo-spin-21 condensates and a nematic vortex lattice in which uniaxial and biaxial spin textures align alternatively for spin-2 condensates. For the pseudo-spin-21 system, although mean-field states always break time-reversal symmetry, there exists a time-reversal invariant many-body ground state, which is fragmented and expected to be observed in a microcondensate.
In this thesis, we study theoretically and in experimentally the properties of spin-1 Bose-Einstein condensates with antiferromagnetic interactions realized in the ultra-cold sodium gases confined in optical dipole traps. We present in detail how to realize, diagnose and control the spinor Bose-Einstein condensates in our experiment. In order to describe the condensate, a mean-field theory is first adopted. This theory predicts a phase transition when changing the magnetic field B. The predictions of the mean-field theory agree very well with most of our experimental results, including the values of the critical magnetic field Bc, asymptotic values of n0 (relative atom number in mF=0 state) at large B and the phase diagram of n0. However, for small magnetization and small magnetic fields, we find abnormally large fluctuations (super-Poissonian) of n0. In order to describe these large fluctuations, we develop a full quantum statistical description for the spinor condensate at finite temperature. To describe uncondensed thermal atoms (also present in the same trap), we use the ``semi-ideal" Hartree-Fock approximation to deal with the interactions between the condensate atoms and the thermal ones. The experimental results lead us to introduce two kinds of temperatures, the ``spin temperature" Ts and the ``kinetic temperature" Tk with Ts≪Tk, characterizing respectively the fluctuaitons of the condensate spin and the thermal gas. We conclude that the system is reaching a quasi-equilibrium states, where different degrees of freedom reach equilibrium by separate mechanisms but where the mutual thermalization does not occur over the lifetime of the cloud.