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Physical Review Letters

Number Fluctuation Dynamics of Atomic Spin Mixing inside a Condensate

Lee Chang, Qiang Zhai, Rong Lü, Li You

Published Aug 21, 2007• 20 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.99.080402

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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 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 $(B\text{\ensuremath{-}})$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 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.
We have observed sub-Poissonian spin correlations generated by collisionally induced spin mixing in a spin-1 Bose-Einstein condensate. We measure a quantum noise reduction of $\ensuremath{-}7\text{ }\text{ }\mathrm{dB}$ ($\ensuremath{-}10\text{ }\text{ }\mathrm{dB}$ corrected for detection noise) below the standard quantum limit for the corresponding coherent spin states. The spin fluctuations are detected as atom number differences in the spin states using fluorescent imaging that achieves a detection noise floor of 8 atoms per spin component for a probe time of $100\text{ }\text{ }\ensuremath{\mu}\mathrm{s}$.
A tomography of many-body quantum states of indistinguishable particles is generally performed by engineering couplings between the involved states and a subsequent counting of the occupation numbers. While precise couplings belong to the standard experimental toolbox, an accurate number counting presents a considerable challenge for both photonic and atomic quantum states. Here we present an application of a number-resolving atom counting [New J. Phys. 23, 113046 (2021)] for the state reconstruction of an atomic coherent spin state. We generate the coherent spin state by driving a Rabi frequency between two hyperfine states of an ultracold Rubidium ensemble.The result is analyzed by a number-resolving fluoresence detection setup. We characterize the fidelity of our detector and show that a negative-valued Wigner function is associated with it. The results offer an exciting perspective for a high-fidelity tomography of entangled states and can be applied for the future demonstration of Heisenberg-limited atom interferometry.
The quantum properties of matter waves, in particular quantum correlations and entanglement are an important frontier in atom optics with applications in quantum metrology and quantum information. In this thesis, we report the first observation of sub-Poissonian fluctuations in the magnetization of a spinor 87Rb condensate. The fluctuations in the magnetization are reduced up to 10 dB below the classical shot noise limit. This relative number squeezing is indicative of the predicted pair-correlations in a spinor condensate and lay the foundation for future experiments involving spin-squeezing and entanglement measurements. We have investigated the limits of the imaging techniques used in our lab, absorption and fluorescence imaging, and have developed the capability to measure atoms numbers with an uncertainly < 10 atoms. Condensates as small as ≈ 10 atoms were imaged and the measured fluctuations agree well with the theoretical predictions. Furthermore, we implement a reliable calibration method of our imaging system based on quantum projection noise measurements. We have resolved the individual lattice sites of a standing-wave potential created by a CO2 laser, which has a lattice spacing of 5.3 µm. Using microwaves, we site-selectively address and manipulate the condensate and therefore demonstrate the ability to perturb the lattice condensate of a local level. Interference between condensates in adjacent lattice sites and lattice sites separated by a lattice site are observed.