Using squeezed states it is possible to surpass the standard quantum limit of measurement uncertainty by reducing the measurement uncertainty of one property at the expense of another complementary property [1]. Squeezed states were first demonstrated in optical fields [2] and later with ensembles of pseudo spin-1/2 atoms using non-linear atom-light interactions [3]. Recently, collisional interactions in ultracold atomic gases have been used to generate a large degree of quadrature spin squeezing in two-component Bose condensates [4,5]. For pseudo spin-1/2 systems, the complementary properties are the different components of the total spin vector S , which fully characterize the state on an SU(2) Bloch sphere. Here, we measure squeezing in a spin-1 Bose condensate, an SU(3) system, which requires measurement of the rank-2 nematic or quadrupole tensor Q ij as well to fully characterize the state. Following a quench through a nematic to ferromagnetic quantum phase transition, squeezing is observed in the variance of the quadratures up to −8.3 +0.6 −0.7 dB (−10.3 +0.7 −0.9 dB corrected for detection noise) below the standard quantum limit. This spin-nematic squeezing is observed for negligible occupation of the squeezed modes and is analogous to optical two-mode vacuum squeezing. This work has potential applications to continuous variable quantum information and quantum-enhanced magnetometry.The study of many-body quantum entangled states including atomic spin squeezed states is an active research frontier. In addition to being intrinsically fascinating, such states have applications in precision measurements [6], quantum information and fundamental tests of quantum mechanics [7]. Atomic squeezed states were first considered for ensembles of two-level (pseudo spin-1/2) atoms. For spin-1/2 particles, coherent states of the system are uniquely specified by the components of the total spin vector S , typically illustrated on a SU(2) Bloch sphere. For particles with higher spin, additional degrees of freedom beyond the spin vector are required to fully specify the state. For spin-1 particles, a natural basis to describe the wavefunction is the SU(3) Cartesian dipole-quadrupole basis, consisting of the three components of the spin vector,Ŝ i , and the moments of the rank-2 quadrupole or nematic tensor,Q ij ({i, j} ∈ {x, y, z}). In matrix form, the nematic moments can be written. Spin-1 atomic Bose-Einstein condensates [9-13] provide an exceptionally clean experimental platform to investigate the quantum dynamics of many-body spin sys-tems. They feature controllable quantum phase transitions, well-understood underlying microscopic models, and flexible defect-free geometries. Importantly, it is possible to initialize non-equilibrium or excited states of the system and to directly measure both the spin vector and the nematic tensor using standard atomic state manipulation tools. Law, et al., demonstrated that the spinor interaction can be written as total spin angular momentum, λŜ 2 whereŜ 2 =Ŝ 2x +Ŝ 2 y +Ŝ 2 z [14]. It is ...
Collisions in a thermal gas are perceived as random or incoherent as a consequence of the large numbers of initial and final quantum states accessible to the system. In a quantum gas, e.g. a Bose-Einstein condensate or a degenerate Fermi gas, the phase space accessible to low energy collisions is so restricted that collisions become coherent and reversible. Here, we report the observation of coherent spin-changing collisions in a gas of spin-1 bosons. Starting with condensates occupying two spin states, a condensate in the third spin state is coherently and reversibly created by atomic collisions. The observed dynamics are analogous to Josephson oscillations in weakly connected superconductors and represent a type of matter-wave four-wave mixing. The spin-dependent scattering length is determined from these oscillations to be -1.45(32) Bohr. Finally, we demonstrate coherent control of the evolution of the system by applying differential phase shifts to the spin states using magnetic fields.PACS numbers:
The combination of cold atoms and large coherent coupling enables investigations in a new regime in cavity QED with single-atom trajectories monitored in real time with high signal-to-noise ratio. The underlying "vacuum-Rabi" splitting is clearly reflected in the frequency dependence of atomic transit signals recorded atom by atom, with evidence for mechanical light forces for intracavity photon number ,1. The nonlinear optical response of one atom in a cavity is observed to be in accord with the one-atom quantum theory but at variance with semiclassical predictions. [S0031-9007(98)06037-2] PACS numbers: 42.50. Ct, 42.50.Vk An important trend in modern physics has been the increasing ability to isolate and manipulate the dynamical processes of individual quantum systems, with interactions studied quantum by quantum. In optical physics, examples include cavity QED with single atoms and photons [1] and trapped ions cooled to the motional zero point [2], while in condensed matter physics, an example is the Coulomb blockade with discrete electron energies [3]. An essential ingredient in these endeavors is that the components of a complex quantum system should interact in a controlled fashion with minimal decoherence. More quantitatively, if the off-diagonal elements of the system's interaction Hamiltonian are characterized by ͗H int ͘ ϳhg, where g is the rate of coherent, reversible evolution, then a necessary requirement is to achieve strong coupling for which g . b ϵ max͓G, T 21 ͔ with T as the interaction time and G as the set of decoherence rates for the system.Although there are many facets to investigations of such open quantum systems, our primary motivation has been to exploit strong coupling in cavity QED to enable research in quantum measurement and more generally, in the emerging field of quantum information dynamics [4]. Several experiments in cavity QED have investigated the nonperturbative interaction of an atom with the electromagnetic field at the level of a single photon; for this system 2g 0 is the single-photon Rabi frequency and G ϵ ͕g Ќ , k͖, with g Ќ as the atomic dipole decay rate and k as the rate of decay of the cavity field [5][6][7][8]. However, without exception these experiments have employed atomic beams in settings for which the information per atomic transit (of duration T ) is I ϵ b ϳ 1, so that measurements over an ensemble of atoms are required. For example, the passage of a Rydberg atom through a microwave cavity and its subsequent measurement provides a single bit of information [5,7].By contrast, an exciting recent development in cavity QED has been the ability to observe single-atom trajectories in real time with I ¿ 1 [9]. In this method the transmitted power of a probe beam is monitored as cold atoms fall between the mirrors of a high-finesse optical resonator, with the probe transmission significantly altered by the position-dependent interaction between atom and cavity field [10,11].Similarly enabled by the use of cold atoms, the research reported in this Letter exploits...
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