We analyze the dynamics of a dilute, trapped Bose-condensed atomic gas coupled to a diatomic molecular Bose gas by coherent Raman transitions. This system is shown to result in a new type of "superchemistry," in which giant collective oscillations between the atomic and the molecular gas can occur. The phenomenon is caused by stimulated emission of bosonic atoms or molecules into their condensate phases.

We calculate the two-particle local correlation for an interacting 1D Bose gas at finite temperature and classify various physical regimes. We present the exact numerical solution by using the Yang-Yang equations and Hellmann-Feynman theorem and develop analytical approaches. Our results draw prospects for identifying the regimes of coherent output of an atom laser, and of finitetemperature "fermionization" through the measurement of the rates of two-body inelastic processes, such as photo-association. The knowledge of the exact solutions to the 1D models allows us to go far beyond the mean-field Bogoliubov approximation. In the current stage of studies of experimentally feasible 1D Bose gases, one of the most important issues that requires such an approach is understanding the correlation properties in the various regimes at finite temperatures.In this Letter we give an exact calculation of the finitetemperature two-particle local correlation for an interacting uniform 1D Bose gas,is the field operator and n = Ψ † (x)Ψ(x) is the linear (1D) density. As a result, we identify and classify various finite-temperature regimes of the 1D Bose gas. Aside from this, the pair correlations are responsible for the rates of inelastic collisional processes [9], and are of particular importance for the studies of coherence properties of atom "lasers" produced in 1D waveguides.At T = 0, the local two-and three-particle correlations of a uniform 1D Bose gas have recently been calculated in Ref. [10]. Here one has two well-known and physically distinct regimes of quantum degeneracy. For weak couplings or high densities, the gas is in a coherent or Gross-Pitaevskii (GP) regime with g (2) → 1. In this regime, long-range order is destroyed by longwavelength phase fluctuations [11] and the equilibrium state is a quasi-condensate characterized by suppressed density fluctuations. For strong couplings or low densities, the gas reaches the strongly interacting or TonksGirardeau (TG) regime and undergoes "fermionization" [3,4]: the wave function strongly decreases as particles approach each other, and g (2) → 0.At the non-zero temperatures studied here, one has to further extend the classification of different regimes. For strong enough couplings or low densities, we obtain the TG regime with g (2) → 0 not only at low temperatures, but also at high temperatures. In addition, and in contrast to previous T = 0 results, we find a weak-coupling or high-density regime in which fluctuations are enhanced. Asymptotically, they reach the non-interacting Bose gas level of g (2) → 2 (rather than g (2) → 1), for any finite temperature T .The emergence of this behavior at low temperatures implies that one can identify three physically distinct regimes of quantum degeneracy: the strong-coupling TG regime of "fermionization" with g (2) → 0, a coherent GP regime with g (2) ≃ 1 at intermediate coupling strength, and a fully decoherent quantum (DQ) regime with g (2) ≃ 2 at very weak couplings. In the GP regime, where the density fluctuations are suppresse...

We investigate the behavior of a weakly interacting nearly one-dimensional (1D) trapped Bose gas at finite temperature. We perform in situ measurements of spatial density profiles and show that they are very well described by a model based on exact solutions obtained using the Yang-Yang thermodynamic formalism, in a regime where other, approximate theoretical approaches fail. We use Bose-gas focusing [Shvarchuck et al., Phys. Rev. Lett. 89, 270404 (2002)] to probe the axial momentum distribution of the gas, and find good agreement with the in situ results.PACS numbers: 03.75. Hh, 05.30.Jp, 05.70.Ce Reducing the dimensionality in a quantum system can have dramatic consequences. For example, the 1D Bose gas with repulsive delta-function interaction exhibits a surprisingly rich variety of physical regimes that is not present in 2D or 3D [1,2]. This 1D Bose gas model is of particular interest because exact solutions for the manybody eigenstates can be obtained using a Bethe ansatz [3]. Furthermore, the finite-temperature equilibrium can be studied using the Yang-Yang thermodynamic formalism [4,5,6], a method also known as the thermodynamic Bethe ansatz. This formalism is the unifying framework for the thermodynamics of a wide range of exactly solvable models. It yields solutions to a number of important interacting many-body quantum systems and as such provides critical benchmarks to condensed-matter physics and field theory [6]. The specific case of the 1D Bose gas as originally solved by Yang and Yang [4] is of particular interest because it is the simplest example of the formalism. The experimental achievement of ultracold atomic Bose gases in the 1D regime [7] has attracted renewed attention to the 1D Bose gas problem [8] and is now providing previously unattainable opportunities to test the Yang-Yang thermodynamics.In this paper, we present the first direct comparison between experiments and theory based on the Yang-Yang exact solutions. The comparison is done in the weakly interacting regime and covers a wide parameter range where conventional models fail to quantitatively describe in situ measured spatial density profiles. Furthermore, we use Bose-gas focusing [9] to probe the equilibrium momentum distribution of the 1D gas, which is difficult to obtain through other means.For a uniform 1D Bose gas, the key parameter is the dimensionless interaction strength γ = mg/ 2 n, where m is the mass of the particles, n is the 1D density, and g is the 1D coupling constant. At low densities or large coupling strength such that γ ≫ 1, the gas is in the strongly interacting or Tonks-Girardeau regime [10]. The opposite limit γ ≪ 1 corresponds to the weakly interacting gas. Here, for temperatures below the degeneracy temperature T d = 2 n 2 /2mk B , one distinguishes two regimes [11]. (i) For sufficiently low temperatures, T ≪ √ γT d , the equilibrium state is a quasi-condensate with suppressed density fluctuations. The system can be treated by the mean-field approach and by the Bogoliubov theory of excitations. The 1D c...

We analyze the coherent formation of molecular Bose-Einstein condensate (BEC) from an atomic BEC, using a parametric field theory approach. We point out the transition between a quantum soliton regime, where atoms couple in a local way to a classical soliton domain, where a stable coupledcondensate soliton can form in three dimensions. This gives the possibility of an intense, stable atomlaser output. [S0031-9007(98) The coherently coupled atom-molecular condensate could provide a route to the observation of a localized threedimensional BEC soliton, even in the absence of a trap potential. A possible application is in the free propagation of a nondiverging atom-laser pulse, thus greatly increasing the intensity in an atom-laser beam. Even more than this would be the importance of observing the striking physical properties of this novel quantum field theory, and the corresponding Bose-enhanced chemical kinetics.The original solution for the parametric soliton was in a one-dimensional environment [6]. These classical solutions have been classified topologically [7], and are generic to the mean-field theories of parametric nonlinearities that convert one particle into two (and vice versa). The equations are nonintegrable, and are different to the usual integrable classes of soliton equations. A considerable advantage of these types of nonlinear equations is that they are capable of providing solutions in one, two, or three space dimensions, which does not occur in the usual Gross-Pitaevskii equations. Both classical [6][7][8] and quantum [9] solutions have been recently identified (including observation of classical solitons in experiment [10]), although these different types of soliton have strikingly different qualitative behavior.The purpose of this Letter is to point out the physical origin of these differences between the quantum and classical versions of the parametric field theory and to identify experimental requirements for observing these novel effects in Bose condensates. We consider the following basic Hamiltonian, to give a simple model of molecule formation:where the free and interacting Hamiltonians arê

We report on local, in situ measurements of atom number fluctuations in slices of a onedimensional Bose gas on an atom chip setup. By using current modulation techniques to prevent cloud fragmentation, we are able to probe the crossover from weak to strong interactions. For weak interactions, fluctuations go continuously from super-to sub-Poissonian as the density is increased, which is a signature of the transition between the sub-regimes where the two-body correlation function is dominated respectively by thermal and quantum contributions. At stronger interactions, the super-Poissonian region disappears, and the fluctuations go directly from Poissonian to subPoissonian, as expected for a 'fermionized' gas.PACS numbers: 03.75. Hh, 67.10.Ba Fluctuations witness the interplay between quantum statistics and interactions and therefore their measurement constitutes an important probe of quantum manybody systems. In particular, measurement of atom number fluctuations in ultracold quantum gases has been a key tool in the study of the Mott insulating phase in optical lattices [1], isothermal compressibility of Bose and Fermi gases [2][3][4][5], magnetic susceptibility of a strongly interacting Fermi gas [6], scale invariance of a twodimensional Bose gas [7], generation of atomic entanglement in double-wells [8], and relative number squeezing in pair-production via binary collisions [9,10].While a simple account of quantum statistics can change the atom number distribution, in a small volume of an ideal gas, from a classical-gas Poissonian to superPoissonian (for bosons) or sub-Poissonian (for fermions) distributions, many-body processes can further modify the correlations and fluctuations. For example, threebody losses may lead to sub-Poissonian fluctuations in a Bose gas [11,12]. Even without dissipation, the intrinsic interatomic interactions can also lead to sub-Poissonian fluctuations, such as in a repulsive Bose gas in a periodic lattice potential, where the energetically costly atom number fluctuations are suppressed. This effect has been observed for large ratios of the on-site interaction energy to the inter-site tunnelling energy [13,14], with the extreme limit corresponding to the Mott insulator phase [15,16]. The same physics, accounts for sub-Poissonian fluctuations observed in double-well experiments [8,17]. Sub-Poissonian fluctuations of the total atom number have been also realised via controlled loading of the atoms into very shallow traps [18].In this work, we observe for the first time subPoissonian atom number fluctuations in small slices of a single one-dimensional (1D) Bose gas with repulsive interactions, where each slice approximates a uniform system. Taking advantage of the long scale density varia- Nearly ideal Bose gasQuasi-condensate tion due to a weak longitudinal confinement, we monitor -at a given temperature -the atom number fluctuations in each slice as a function of the local density. For a weakly interacting gas, the measured fluctuations are super-Poissonian at low densities, and t...

We experimentally demonstrate a nonlinear detection scheme exploiting time-reversal dynamics that disentangles continuous variable entangled states for feasible readout. Spin-exchange dynamics of Bose-Einstein condensates is used as the nonlinear mechanism which not only generates entangled states but can also be time reversed by controlled phase imprinting. For demonstration of a quantumenhanced measurement we construct an active atom SU(1,1) interferometer, where entangled state preparation and nonlinear readout both consist of parametric amplification. This scheme is capable of exhausting the quantum resource by detecting solely mean atom numbers. Controlled nonlinear transformations widen the spectrum of useful entangled states for applied quantum technologies.Nonlinear dynamics is the basis of generating nonclassical states of many particles. These entangled states are capable of improving a large variety of operations, e.g., computational tasks [1], communication and measurements [2]. Unlocking their full potential for quantum technologies requires both the generation and detection at the fundamental quantum limit. The generation of such highly entangled states with many particles has witnessed tremendous advances [3,4]. However, to fully exploit this quantum resource, the complete correlations on the single particle level need to be accessed, which still limits current experiments.To address this challenge, nonlinear readout schemes have been proposed [5][6][7][8]. Most of these employ a time inversion sequence. For this the nonlinear evolution that is used to produce the entangled state is inverted and reapplied for readout. If the state remains unperturbed, the second period of nonlinear evolution counteracts the first. This time-reversed readout disentangles the probe state such that the known separable initial state is recovered. This reversibility is nonperfect if the state is changed in between, similar to an incomplete Loschmidt-Echo [9]. By this sensitive mechanism, minute state perturbations are mapped onto readily discernable quantities.Experimentally, we use spin-changing collisions [10] in a mesoscopic spinor Bose-Einstein condensate. This nonlinear mechanism is the atomic analogue of parametric amplification, which is the textbook example of entangled state generation in quantum optics. At the same time, both the sign and the strength of the nonlinear coupling are experimentally adjustable, which makes this system ideally suited for realizing time reversal readout schemes.Spin exchange is performed in an effective three-level system within the spin F = 2 manifold of 87 Rb. For this the external degrees of freedom are frozen out such that dynamics is restricted to the spin degree of freedom. We start with a pure |F = 2, m F = 0 condensate (pump mode). Population in any m F = 0 state is carefully cleaned. During spin mixing atoms of the pump mode are coherently and pairwise scattered into the signal |↑ ≡ |2, 1 and idler |↓ ≡ |2, −1 mode, which we refer to as side modes (see Fig. 1). For small p...

The Cauchy-Schwarz (CS) inequality -one of the most widely used and important inequalities in mathematics -can be formulated as an upper bound to the strength of correlations between classically fluctuating quantities. Quantum-mechanical correlations can, however, exceed classical bounds. Here we realize four-wave mixing of atomic matter waves using colliding Bose-Einstein condensates, and demonstrate the violation of a multimode CS inequality for atom number correlations in opposite zones of the collision halo. The correlated atoms have large spatial separations and therefore open new opportunities for extending fundamental quantum-nonlocality tests to ensembles of massive particles. 03.75.Gg, 34.50.Cx, 42.50.Dv The Cauchy-Schwarz (CS) inequality is ubiquitous in mathematics and physics [1]. Its utility ranges from proofs of basic theorems in linear algebra to the derivation of the Heisenberg uncertainty principle. In its basic form, the CS inequality simply states that the absolute value of the inner product of two vectors cannot be larger than the product of their lengths. In probability theory and classical physics the CS inequality can be applied to fluctuating quantities and states that the expectation value of the cross-correlation I 1 I 2 between two quantities I 1 and I 2 is bounded from above by the auto-correlations in each quantity:This inequality is satisfied, for example, by two classical currents emanating from a common source. In quantum mechanics, correlations can, however, be stronger than those allowed by the CS inequality [2][3][4]. Such correlations have been demonstrated in quantum optics using, for example, antibunched photons produced via spontaneous emission [5], or twin photon beams generated in a radiative cascade [6], parametric down conversion [7], and optical fourwave mixing [8]. Here the discrete nature of the light and the strong correlation (or anticorrelation in antibunching) between photons is responsible for the violation of the CS inequality. The violation has even been demonstrated for two light beams detected as continuous variables [8].In this work we demonstrate a violation of the CS inequality in matter-wave optics using pair-correlated atoms formed in a collision of two Bose-Einstein condensates (BECs) of metastable helium [9-12] (see Fig. 1). The CS inequality which we study is a multimode inequality, involving integrated atomic densities, and therefore is different from the typical two-mode situation studied in quantum optics. Our results demonstrate the potential of atom optics experiments to extend the fundamental tests of quantum mechanics to ensem- Spherical halo of scattered atoms produced by four-wave mixing after the cloud expands and the atoms fall to the detector 46 cm below. During the flight to the detector, the unscattered condensates acquire a disk shape shown in white on the north and south poles of the halo. The (red) boxes 1 and 2 illustrate a pair of diametrically symmetric counting zones (integration volumes) for the average cross-and autocorrela...

We calculate the density profiles and density correlation functions of the one-dimensional Bose gas in a harmonic trap, using the exact finite-temperature solutions for the uniform case, and applying a local density approximation. The results are valid for a trapping potential that is slowly varying relative to a correlation length. They allow a direct experimental test of the transition from the weak-coupling Gross-Pitaevskii regime to the strong-coupling, "fermionic" Tonks-Girardeau regime. We also calculate the average two-particle correlation which characterizes the bulk properties of the sample, and find that it can be well approximated by the value of the local pair correlation in the trap center.