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Physical Review A

Number fluctuation dynamics of a spin-1 atomic condensate

Qiang Zhai, Lee Chang, Rong Lü, Li You

Published Apr 10, 2009• 5 citations
Publisher

Abstract

We provide further details on atom number fluctuation during spin-mixing dynamics inside an $F=1$ spinor condensate, based on the effective quantization approach we reported earlier [L. Chang et al., Phys. Rev. Lett. 99, 080402 (2007)] for the semiclassical nonrigid pendulum model in terms of the relative phase and the atomic population in the magnetic $(B\text{\ensuremath{-}})$field insensitive hyperfine state $|{F}_{z}=0⟩$. Many features of the classical nonrigid pendulum model are reproduced. In addition, features typically associated with full quantum simulations specific to our model system, such as atom number fluctuations both in stationary states and during the dynamics of spin mixing, are faithfully revealed with our effective quantization approach. We check the validity of the effective quantization theory by performing extensive numerical comparisons with the results from full quantum many-body simulations. We study the effect of quantum coherence in our system and provide a solid theoretical basis to resolve the resolution limit in counting atoms that is needed to clearly detect quantum correlation effects in spin mixing.

DOI
10.1103/physreva.79.043608

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We provide further details on atom number fluctuation during spin-mixing dynamics inside an F=1F=1F=1 spinor condensate, based on the effective quantization approach we reported earlier [L. Chang et al., Phys. Rev. Lett. 99, 080402 (2007)] for the semiclassical nonrigid pendulum model in terms of the relative phase and the atomic population in the magnetic (B\ensuremath-)(B\text{\ensuremath{-}})(B\ensuremath-)field insensitive hyperfine state ∣Fz=0⟩|{F}_{z}=0⟩∣Fz​=0⟩. Many features of the classical nonrigid pendulum model are reproduced. In addition, features typically associated with full quantum simulations specific to our model system, such as atom number fluctuations both in stationary states and during the dynamics of spin mixing, are faithfully revealed with our effective quantization approach. We check the validity of the effective quantization theory by performing extensive numerical comparisons with the results from full quantum many-body simulations. We study the effect of quantum coherence in our system and provide a solid theoretical basis to resolve the resolution limit in counting atoms that is needed to clearly detect quantum correlation effects in spin mixing.
We investigate the quantum dynamics of number fluctuations inside an atomic condensate during coherent spin mixing among internal states of the ground state hyperfine manifold, by quantizing the semiclassical nonrigid pendulum model in terms of the conjugate variable pair: the relative phase and the atom number. Our result provides a theoretical basis that resolves the resolution limit, or the effective "shot-noise" level, for counting atoms that is needed to clearly detect quantum correlation effects in spin mixing.
We study the coherent off-equilibrium spin mixing inside an atomic condensate. Using mean-field theory and adopting the single-spatial-mode approximation, the condensate spin dynamics is found to be well described by that of a nonrigid pendulum and displays a variety of periodic oscillations in an external magnetic field. Our results illuminate several recent experimental observations and provide critical insights into the observation of coherent interaction-driven oscillations in a spin-1 condensate.
We study quantum spin mixing in a mixture of two spin-1 condensates including coherent interspecies mixing processes, using condensates of 87{}^{87}87Rb and 23{}^{23}23Na atoms in the ground lower hyperfine F=1F=1F=1 manifold as prototype examples. Adopting the single spatial mode approximation for each of the two spinor condensates, we find the mixing dynamics reduce to that of three coupled nonrigid pendulums with clear physical interpretation. With suitably prepared initial states, the spin mixing dynamics allow for the determination of interspecies singlet-pairing as well as spin-exchange interaction parameters.
The primary study of this thesis is the experimental realization of the non-equilibrium dynamics of a quantum inverted pendulum as examined in the collective spin dynamics of a spin-1 Bose-Einstein condensate. In order to compare experimental results with the simulation past the low depletion limit, current simulation techniques needed to be extended to model atomic loss. These extensions show that traditional measurements of the system evolution (e.g. measuring the mean and standard deviation of the evolving quantity) were insufficient in capturing the quantum nature of the evolution. It became necessary to look at higher order moments and cumulants of the distributions in order to capture the quantum fluctuations. Extending the implications of the loss model further, it is possible that the system evolves in a way previously unpredicted. Spin-mixing from a hyperbolic fixed point in the phase space and low noise atom counting form the core of the experiment to measure the evolution of the distributions of the spin populations. The evolution of the system is also compared to its classical analogue, the momentum-shortened inverted pendulum. \n \nThe other experimental study in this thesis is mapping the mean-field phase space. The mean-field phase space consists of different energy contours that are divided into both phase-winding trajectories and closed orbits. These two regions are divided by a separatrix whose orbit has infinite period. Coherent states can be created fairly accurately within the phase space and allowed to evolve freely. The nature of their subsequent evolution provides the shape of the phase space orbit at that initial condition. From this analysis a prediction of the nature of the entire phase space is possible.
Decoherence with recurrences appear in the dynamics of the one-body density matrix of an F=1 spinor Bose–Einstein condensate, initially prepared in coherent states, in the presence of an external uniform magnetic field and within the single mode approximation. The phenomenon emerges as a many-body effect of the interplay of the quadratic Zeeman effect, which breaks the rotational symmetry, and the spin-spin interactions. By performing full quantum diagonalizations, a very accurate time evolution of large condensates is analyzed, leading to heuristic analytic expressions for the time dependence of the one-body density matrix, in the weak and strong interacting regimes, for initial coherent states. We are able to find accurate analytical expressions for both the decoherence and the recurrence times, in terms of the number of atoms and strength parameters, which show remarkable differences depending on the strength of the spin-spin interactions. The features of the stationary states in both regimes are also investigated. We discuss the nature of these limits in light of the thermodynamic limit.
We study particle number fluctuations in the quantum ground states of a mixture of two spin-1 atomic condensates when the interspecies spin-exchange coupling interaction ${c}_{12}\ensuremath{\beta}$ is adjusted. The two spin-1 condensates forming the mixture are, respectively, ferromagnetic and polar in the absence of an external magnetic (B) field. We categorize all possible ground states using the angular momentum algebra and compute their characteristic atom number fluctuations, focusing especially on the the AA phase (when ${c}_{12}\ensuremath{\beta}>0$), where the ground state becomes fragmented and atomic-number fluctuations exhibit drastically different features from a single standalone spin-1 polar condensate. Our results are further supported by numerical simulations of the full quantum many-body system.