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arXiv:0710.3547 · Physical Review A

Quantum noise, scaling, and domain formation in a spinor Bose-Einstein condensate

George I. Mias, Nigel R. Cooper, S. M. Girvin

Published Feb 14, 2008• 44 citations
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Abstract

In this paper we discuss Bose-Einstein spinor condensates for $F=1$ atoms in the context of $^{87}\text{R}\text{b}$, as studied experimentally by the Stamper-Kurn group [L. E. Sadler et al., Nature (London) 443, 312 (2006)]. The dynamical quantum fluctuations of a sample that starts as a condensate of $N$ atoms in a pure $F=1$, ${m}_{F}=0$ state are described in analogy to the two-mode squeezing of quantum optics in terms of an $\mathfrak{s}\mathfrak{u}(1,1)$ algebra. In this system the initial ${m}_{F}=0$ condensate acts as a source (pump) for the creation pairs of ${m}_{F}=1,\ensuremath{-}1$ atoms. We show that even though the system as a whole is described by a pure state with zero entropy, the reduced density matrix for the ${m}_{F}=+1$ degree of freedom, obtained by tracing out the ${m}_{F}=\ensuremath{-}1,0$ degrees of freedom, corresponds to a thermal state. Furthermore, these quantum fluctuations of the initial dynamics of the system provide the seeds for the formation of domains of ferromagnetically aligned spins.

DOI
10.1103/physreva.77.023616

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In this paper we discuss Bose-Einstein spinor condensates for F=1F=1F=1 atoms in the context of 87Rb^{87}\text{R}\text{b}87Rb, as studied experimentally by the Stamper-Kurn group [L. E. Sadler et al., Nature (London) 443, 312 (2006)]. The dynamical quantum fluctuations of a sample that starts as a condensate of NNN atoms in a pure F=1F=1F=1, mF=0{m}_{F}=0mF​=0 state are described in analogy to the two-mode squeezing of quantum optics in terms of an su(1,1)\mathfrak{s}\mathfrak{u}(1,1)su(1,1) algebra. In this system the initial mF=0{m}_{F}=0mF​=0 condensate acts as a source (pump) for the creation pairs of mF=1,\ensuremath−1{m}_{F}=1,\ensuremath{-}1mF​=1,\ensuremath−1 atoms. We show that even though the system as a whole is described by a pure state with zero entropy, the reduced density matrix for the mF=+1{m}_{F}=+1mF​=+1 degree of freedom, obtained by tracing out the mF=\ensuremath−1,0{m}_{F}=\ensuremath{-}1,0mF​=\ensuremath−1,0 degrees of freedom, corresponds to a thermal state. Furthermore, these quantum fluctuations of the initial dynamics of the system provide the seeds for the formation of domains of ferromagnetically aligned spins.
Recent experimental studies of 87Rb^{87}\text{R}\text{b}87Rb spinor Bose Einstein condensates have shown the existence of a magnetic field driven quantum phase transition accompanied by structure formation on the ferromagnetic side of the transition. In this theoretical study we examine the dynamics of the unstable modes following the transition within the framework of the semiclassical truncated Wigner approximation. In the process we present a systematic study of the effects of the trap, nonlinearities, finite temperature, and dipole-dipole interactions. Starting from an initial state which includes quantum fluctuations, we attempt to make quantitative comparisons with recent experimental data on this system by combining results presented here with those presented previously [S. Leslie, J. Guzman, M. Vengalattore, J. D. Sau, M. L. Cohen, and D. M. Stamper-Kurn, Phys. Rev. A 79, 043631 (2009)]. In the process we estimate the contribution of quantum zero-point fluctuations to the domain formation with quantitative accuracy and find discrepancies between the calculations and experiments at the quantitative level, despite the qualitative agreement between theory and experiment. We discuss the possible origins of these discrepancies. Finally, using the strong anisotropy of the trap, we propose ways to observe directly the effects of dipole-dipole interactions on the spinor condensate dynamics.
Recent experimental studies of 87^{87}87Rb spinor Bose Einstein condensates have shown the existence of a magnetic field driven quantum phase transition accompanied by structure formation on the ferromagnetic side of the transition. In this theoretical study we examine the dynamics of the unstable modes following the transition taking into account the effects of the trap, non-linearities, finite temperature and dipole-dipole interactions. Starting from an initial state which includes quantum fluctuations, we attempt to make quantitative comparisons with recent experimental data on this system and estimate the contribution of quantum zero-point fluctuations to the domain formation. Finally, using the strong anisotropy of the trap, we propose ways to observe directly the effects of dipole-dipole interactions on the spinor condensate dynamics.
Motivated by a recent experiment, we study nonequilibrium quantum phenomena taking place in the quench of a spinor Bose-Einstein condensate through the zero-temperature phase transition separating the polar paramagnetic and planar ferromagnetic phases. We derive the typical spin domain structure (correlations of the effective magnetization) created by the quench arising due to spin-mode quantum fluctuations, and we establish a sample-size scaling law for the creation of spin vortices, which are topological defects in the transverse magnetization.
Bose-Einstein condensation (BEC) is a phenomenon in which identical bosons occupy the same quantum state below a certain critical temperature. A hallmark of BEC is the coherence between particles every particle shares the same quantum wavefunction and phase. This coherence has been demonstrated for the external (motional) degrees of freedom of the atomic condensates by interfering two condensates. In this thesis, the coherence is shown to extend to the internal spin degrees of freedom of a spin-1 Bose gas evidenced by the observed coherent and reversible spin-changing collisions. The observed coherent dynamics are analogous to Josephson oscillations in weakly connected superconductors and represent a type of matter-wave four-wave mixing. Control of the coherent evolution of the system using magnetic fields is also demonstrated. The studies on spinor condensates begin by creating spinor condensates directly using all-optical approaches that were first developed in our laboratory. All-optical formation of Bose-Einstein condensates (BEC) in 1D optical lattice and single focus trap geometries are developed and presented. These techniques offer considerable flexibility and speed compared to magnetic trap approaches, and the trapping potential can be essentially spin-independent and are ideally suited for studying spinor condensates. Using condensates with well-defined initial non-equilibrium spin configuration, spin mixing of F = 1 and F = 2 spinor condensates of rubidium-87 atoms confined in an optical trap is observed. The equilibrium spin configuration in the F = 1 manifold confirms that 87Rb is ferromagnetic. The coherent spinor dynamics are demonstrated by initiating spin mixing deterministically with a non-stationary spin population configuration. Finally, the interplay between the coherent spin mixing and spatial dynamics in spin-1 condensates with ferromagnetic interactions is investigated.
We explore the time evolution of two-component Bose-Einstein condensates (BECs), quasi one dimensional with respect to their spinor dynamics, following a quench from one component BECs with a U(1)\mathrm{U}(1)U(1) order parameter into two-component condensates with a $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ order parameter. In our case, these two spin components have a propensity to phase separate, i.e., they are immiscible. Remarkably, these spin degrees of freedom can equivalently be described as a single-component attractive BEC. A spatially uniform mixture of these spins is dynamically unstable, rapidly amplifying any quantum or pre-existing classical spin fluctuations. This coherent growth process drives the formation of numerous spin-polarized domains, which are far from the system's ground state. At much longer times these domains grow in size, coarsening, as the system approaches equilibrium. The experimentally observed time evolution is consistent with our fully three-dimensional stochastic-projected Gross--Pitaevskii calculation.
We discuss the quantum phases and their diffusion in a spinor-1 atomic Bose-Einstein condensate. For ferromagnetic interactions, we obtain the exact ground state distribution of the phase fluctuations corresponding to the total atom number (N), the magnetization (M), and the alignment (or hypercharge) (Y) of the system. The mean-field ground state is shown to be stable against these fluctuations, which dynamically recover the two continuous symmetries associated with the conservation of N and M as in current experiments.