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 ...
We measure spin mixing of F=1 and F=2 spinor condensates of 87 Rb atoms confined in an optical trap. We determine the spin mixing time to be typically less than 600 ms and observe spin population oscillations. The equilibrium spin configuration in the F=1 manifold is measured for different magnetic fields and found to show ferromagnetic behavior for low field gradients. An F=2 condensate is created by microwave excitation from F=1 manifold, and this spin-2 condensate is observed to decay exponentially with time constant 250 ms. Despite the short lifetime in the F=2 manifold, spin mixing of the condensate is observed within 50 ms.PACS numbers: 03.75. Mn, 32.80.Pj, One of the hallmarks of Bose-Einstein condensation (BEC) in dilute atomic gases is the relatively weak and well-characterized inter-atomic interactions that allow quantitative comparison with theory. The vast majority of experimental work has involved single component systems, using magnetic traps confining just one Zeeman sub-level in the ground state hyperfine manifold. An important frontier in BEC research is the extension to multi-component systems, which provides a unique opportunity for exploring coupled, interacting quantum fluids. In particular, atomic BECs with internal spin degrees of freedom offer a new form of coherent matter with complex internal quantum structures. The first twocomponent condensate was produced utilizing two hyperfine states in 87 Rb, and remarkable phenomena such as phase separation were observed [1,2]. Sodium F=1 spinor BECs have been created by transferring spin polarized condensates into a far-off resonant optical trap to liberate the internal spin degrees of freedom. This allowed investigations of the ground state properties of Na spinor condensates, and observations of domain structures, metastability, and quantum spin tunneling [3,4,5].In this letter, we explore the spin dynamics and ground state properties of 87 Rb spinor condensates in an alloptical trap, by starting with well-characterized initial conditions in a known magnetic field. We focus on the F=1 case and confirm the predicted ferromagnetic behavior. We observe population oscillation between different spin states during the spin mixing and observe reduced magnetization fluctuations, pointing the way to future exploration of the underlying spin squeezing and spin entanglement predicted for the system [6]. We also create F=2 spinors using a microwave excitation, measure a decay of the condensate with a time constant of 250 ms. Despite the short lifetime, spin mixing of the spin-2 condensates is observed within 50 ms. Similar results are concurrently reported in Ref [7]; in that work, the emphasis is on the F=2 mixing, while here, we focus mainly on the F=1 manifold.A spinor BEC can be described by a multi-component order parameter which is invariant under gauge transformation and rotation in spin space [8,9,10]. For a spin-1 BEC, the condensate is either ferromagnetic or antiferromagnetic [8], and the corresponding ground state structure and dynamical prope...
Ultracold 87 Rb atoms are delivered into a high-finesse optical micro-cavity using a translating optical lattice trap and detected via the cavity field. The atoms are loaded into an optical lattice from a magneto-optic trap (MOT) and transported 1.5 cm into the cavity. Our cavity satisfies the strong-coupling requirements for a single intracavity atom, thus permitting real-time observation of single atoms transported into the cavity. This transport scheme enables us to vary the number of intracavity atoms from 1 to >100 corresponding to a maximum atomic cooperativity parameter of 5400, the highest value ever achieved in an atom-cavity system. When many atoms are loaded into the cavity, optical bistability is directly measured in real-time cavity transmission.Many applications in quantum information science require the coherent and reversible interaction of single photon fields with material qubits such as trapped atoms. Quantum states can be transferred between light and matter-respectively offering long range communication and long-term storage of quantum information. This important paradigm is the heart of cavity QED systems, which are largely focused on creating laboratory systems capable of reversible matter-photon dynamics at the single photon level [1]. To achieve this, a small high-finesse build-up cavity is used to tremendously enhance the electric field per photon and hence the interaction strength of a single photon with the cavity medium (e.g. atoms). For a single atom in the cavity, the interaction strength is given by the single photon Rabi frequency, 2g 0 , and coherent dynamics is achieved for g 2 0 /(κΓ) ≫ 1, where κ is the the cavity field decay rate and Γ is the atomic spontaneous emission rate.There have been spectacular recent successes in cavity QED research brought about by the merging of optical cavity systems with ultracold neutral atoms [2], including real-time observation [3,4,5] and trapping [6,7,8,9] of single atoms in optical cavities, real-time feedback control on a single atom [10], and single photon generation [11,12]. Together with the remarkable experimental work in microwave cavity QED [13] and the future prospects for cavity QED with trapped ions [14,15], the field is well-poised to contribute significantly to the development of quantum information science. Indeed, current cavity QED parameters are sufficient for existing quantum gate protocols with fidelities > 99.9% percent [16,17,18,19], and the systems are principally limited by the lack of a scalable atomic trapping system to provide adequate control over atom motional degrees of freedom.Our strategy for overcoming this limitation is to employ optical dipole trapping fields independent from the cavity and orthogonal to its axis as illustrated in Fig.
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 −7 dB (−10 dB corrected for detection noise) below the standard quantum limit (SQL) 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 µs.PACS numbers: 42.65. Hw, 42.50.Dv, 03.75.Gg, 03.75.Mn The study of quantum correlated states including squeezed and entangled states is an active research frontier with important applications in precision measurements, quantum information and fundamental tests of quantum mechanics. Much of the early research in this area focused on quantum optical systems [1], motivated originally by the suggestion that squeezed states could be used in gravity wave detectors to surpass the standard quantum limit [2,3]. There has also been significant progress in realizing squeezing and other quantum correlated (nonclassical) states in atomic systems, using either non-linear atom-light interactions [4], or more recently, collisional interactions in ultracold atomic gases [5][6][7][8][9][10][11][12][13][14][15][16].In optics, squeezed states of light can be generated using non-linear optical interactions that create quantum correlations between different field modes. An important example is optical four-wave mixing, which is a thirdorder parametric process employed in the first demonstration of squeezed states of light in the pioneering experiments by Slusher et al. 25 years ago [17]. In spontaneous four-wave mixing, a strong pump field interacting with medium with a χ (3) non-linearity generates two correlated optical beams known as the signal and idler modes that are exactly correlated in photon number, anticorrelated in phase and exhibit two-mode quadrature squeezing [18].In ultracold atomic gases, binary s-wave collisions between atoms naturally give rise to strong third-order non-linear interactions capable of producing analogous four-wave mixing of atomic matter waves. Indeed, both stimulated and spontaneous atomic four-wave mixing have been observed with colliding condensates [19][20][21][22], and in the spin dynamics of spinor condensates [23][24][25][26][27][28]. Recently, sub-Poissonian correlations were observed in spontaneous four-wave mixing of two colliding condensates [15], manifest as −0.5 dB relative atom number squeezing measured between outgoing modes of opposing momenta in the s-wave scattering halo. The focus of this work is the demonstration of sub-Poissonian spin correlations generated by four-wave spin mixing (4WSM).In spinor condensates, the spin dependence of the collisional interaction gives rise to spin-mixing of the internal states of the matter wave [29][30][31]. For a spin-1 condensate, the mixing of the 3 internal states is described by the interaction Hamiltonian:whereâ i is the annihilation operator...
A pendulum prepared perfectly inverted and motionless is a prototype of unstable equilibrium and corresponds to an unstable hyperbolic fixed point in the dynamical phase space. Here, we measure the non-equilibrium dynamics of a spin-1 Bose-Einstein condensate initialized as a minimum uncertainty spin-nematic state to a hyperbolic fixed point of the phase space. Quantum fluctuations lead to non-linear spin evolution along a separatrix and non-Gaussian probability distributions that are measured to be in good agreement with exact quantum calculations up to 0.25 s. At longer times, atomic loss due to the finite lifetime of the condensate leads to larger spin oscillation amplitudes, as orbits depart from the separatrix. This demonstrates how decoherence of a many-body system can result in apparent coherent behaviour. This experiment provides new avenues for studying macroscopic spin systems in the quantum limit and for investigations of important topics in non-equilibrium quantum dynamics.
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