Entanglement, a key feature of quantum mechanics, is a resource that allows the improvement of precision measurements beyond the conventional bound reachable by classical means [1]. This is known as the standard quantum limit, already defining the accuracy of the best available sensors for various quantities such as time [2] or position [3,4]. Many of these sensors are interferometers in which the standard quantum limit can be overcome by feeding their two input ports with quantum-entangled states, in particular spin squeezed states [5,6]. For atomic interferometers, Bose-Einstein condensates of ultracold atoms are considered good candidates to provide such states involving a large number of particles [7]. In this letter, we demonstrate their experimental realization by splitting a condensate in a few parts using a lattice potential. Site resolved detection of the atoms allows the measurement of the conjugated variables atom number difference and relative phase. The observed fluctuations imply entanglement between the particles [7,8,9], a resource that would allow a precision gain of 3.8 dB over the standard quantum limit for interferometric measurements.Spin squeezing was one of the first quantum strategies proposed to overcome the standard quantum limit in a precision measurement [5,6] that triggered many experiments [10,11,12,13,14,15,16,17]. It applies to measurements where the final readout is done by counting the occupancy difference between two quantum states, as in interferometry or in spectroscopy. The name "spin squeezing" originates from the fact that the N particles used in the measurement can be described by a fictitious spin J = N/2. In an interferometric sequence, the spin undergoes a series of rotations where one of the rotation angles is the phase shift to be measured. A sufficient criterion for the input state allowing for quantum enhanced metrology is given by ξ S < 1 where ξis the squeezing parameter introduced in ref. [6]. The fluctuations of the spin in one direction have to be reduced below shot-noise ∆J 2 z < J/2, and the spin polarization in the orthogonal plane J x 2 + J y 2 has to be large enough to maintain the sensitivity of the interferometer. A pictorial representation of this condition is shown in figure 1b. The precision of such a quantum enhanced measurement is ξ S / √ N , whereas the standard quantum limit set by shot-noise is 1/ √ N . In this Letter, we report on the observation of entangled squeezed states in a Bose-Einstein condensate of 87 Rb atoms. The particles are distributed over a small number of lattice sites (between 2 and 6) in a one dimensional optical lattice (see figure 1a). The occupation number per site ranges from 100 to 1100 atoms. The two modes supporting the squeezing are two states of the external atomic motion corresponding to the condensate meanfield wave-functions in two adjacent lattice sites. These modes are spatially well separated and thus represent an ideal starting condition for a spatially split interferometer. Labeling a † and b † the creation o...

We report on the generation, subsequent oscillation and interaction of a pair of matter wave dark solitons. These are created by releasing a Bose-Einstein condensate from a double well potential into a harmonic trap in the crossover regime between one dimension (1D) and three dimensions (3D). Multiple oscillations and collisions of the solitons are observed, in quantitative agreement with simulations of the Gross-Pitaevskii equation. An effective particle picture is developed and confirms that the deviation of the observed oscillation frequencies from the asymptotic prediction νz/ √ 2, where νz is the longitudinal trapping frequency, results from the dimensionality of the system and the soliton interactions.

Spectral induced polarization as well as complex electrical measurements are used to estimate, on a non‐invasive basis, hydraulic permeability in aquifers. Basic laboratory measurements on a variety of shaly sands, silts and clays showed that the main feature of their conductivity spectra in the frequency range from 10‐3 to 103 Hertz is a nearly constant phase angle. Thus, a constant‐phase‐angle model of electrical conductivity is applied to interpret quantitatively surface and borehole spectral induced polarization measurements. The model allows for the calculation of two independent electrical parameters from only one frequency scan and a simple separation of electrical volume and interface effects. The proposed interpretation algorithm yields the true formation factor, the cation exchange capacity and the surface‐area‐to‐porosity ratio, which corresponds to the inverse hydraulic radius. Using a Kozeny–Carman‐like equation, the estimation of hydraulic permeability is possible.

Petrophysical interpretation of resistivity measurements is often hindered by the dependence of resistivity on the interconnected pore fluids and the interconnected pore surfaces. Induced polarization (IP) measurements yield parameters that are only controlled by the interconnected pore surfaces, thereby offering the opportunity to constrain interpretation of resistivity measurements. Using a database composed of 63 sandstone and unconsolidated sediment samples covering nine independent investigations, we identified a strong linear relationship between the real part of surface conductivity ([Formula: see text]) determined from multisalinity ([Formula: see text]) resistivity measurements and the imaginary conductivity ([Formula: see text]) measured with IP at a frequency of about 1 Hz. We found [Formula: see text] with a coefficient of determination ([Formula: see text]) of 0.911 and a standard deviation of [Formula: see text] of 0.022. We found a similar relation when the normalized chargeability (from Debye decomposition) of the frequency dependence of the IP response is used instead of [Formula: see text]. By estimating the true formation factor ([Formula: see text]) recorded at high salinity, we solved for [Formula: see text] and found that it parallels the salinity dependency of the imaginary conductivity, [Formula: see text], as reported in recent studies. We also found that the value of the [Formula: see text] determined from this experimental study was generally consistent with predictions of the POLARIS model when the mobility of the ions in the Stern layer was assumed to be [Formula: see text] of the mobility of the ions in the diffuse layer (considered equal to the mobility of the ions in the bulk solution). We discovered how the identified relationship can be used to significantly improve (1) the estimation of the true formation factor and (2) the groundwater conductivity, from a single salinity resistivity measurement when an IP measurement is also made. The approach offers an opportunity to improve estimation of porosity, formation factor, and salinity in well logging and hydrogeophysical investigations.

We consider the stability and dynamics of multiple dark solitons in cigar-shaped Bose-Einstein condensates (BECs). Our study is motivated by the fact that multiple matter-wave dark solitons may naturally form in such settings as per our recent work [Phys. Rev. Lett. 101, 130401 (2008)]. First, we study the dark soliton interactions and show that the dynamics of well-separated solitons (i.e., ones that undergo a collision with relatively low velocities) can be analyzed by means of particle-like equations of motion. The latter take into regard the repulsion between solitons (via an effective repulsive potential) and the confinement and dimensionality of the system (via an effective parabolic trap for each soliton). Next, based on the fact that stationary, well-separated dark multi-soliton states emerge as a nonlinear continuation of the appropriate excited eigensates of the quantum harmonic oscillator, we use a Bogoliubov-de Gennes analysis to systematically study the stability of such structures. We find that for a sufficiently large number of atoms, multiple soliton states may be dynamically stable, while for a small number of atoms, we predict a dynamical instability emerging from resonance effects between the eigenfrequencies of the soliton modes and the intrinsic excitation frequencies of the condensate. Finally we present experimental realizations of multi-soliton states including a three-soliton state consisting of two solitons oscillating around a stationary one.Comment: 17 pages, 11 figure

We analyze the relationship between induced polarization ͑IP͒ parameters and the specific surface area normalized to the pore volume ͑S por ͒ for an extensive sample database. We find that a single linear imaginary conductivity-S por relation holds across a range of single-frequency IP data sets composed of sandstones and unconsolidated sediments that lack an appreciable metallic mineral content. We also apply a recent approach defined as Debye decomposition ͑DD͒ to determine normalized chargeability ͑m n ͒, a global estimate of polarization magnitude from available spectral IP ͑SIP͒ data sets. A strong linear relationship between m n and S por is also found across multiple data sets. However, SIP model parameters determined for samples containing metallic minerals are approximately two orders of magnitude greater than for the model parameters estimated for the nonmetallic sample database. We propose a concept of "polarizability of the mineralfluid interface per unit S por " to explain this difference, which is supported by the observed dependence of IP parameters on fluid conductivity between sample types. We suggest that this linear IP-S por relation can be considered the IP equivalent of the classical Archie empirical relation. Whereas the Archie relation describes a power-law relation between electrical conductivity due to electrolytic conduction through the available interconnected pore volume, the IP-S por relation is an equivalent relation between mineral-fluid interfacial polarization and available pore surface area.

We have compared various induced polarization (IP) models for permeability prediction of the same general form that were all based on two parameters, the first being an electric substitute of effective porosity (the formation factor) and the second being an electric proxy of pore-normalized surface area (the imaginary part of electric conductivity). These models (empirically derived and based on mechanistic formulations) were applied to an extensive database acquired on sandstones and unconsolidated sandy materials. Whereas previous studies on permeability prediction mainly concentrated on either sandstone or unconsolidated sediments, we investigated a database composed of 94 samples including sandstones and unconsolidated material. Most of the samples in the database were saturated with a NaCl solution with an electric conductivity close to [Formula: see text]. Samples with a saturating fluid that deviated from this composition were corrected using recently published relationships describing the IP dependence on the pore fluid composition. In the case of the sandstone samples, the electric formation factor exerts the primary control on permeability, and the imaginary conductivity was found to be of little importance in permeability prediction. The opposite was observed for the unconsolidated samples, in which the imaginary conductivity was the most important term for permeability and the formation factor was found to be of little importance. The findings suggest that only one property (formation factor in the case of sandstone, imaginary conductivity in the case of unconsolidated samples) might be needed for the order of magnitude estimates of permeability from the popular form of the model based on the IP observations examined here. Whereas the formation factor was challenging to reliably estimate in situ, the imaginary conductivity was directly obtainable from an IP measurement. This suggests that field scale, single-frequency, IP-based estimation of permeability would be challenging, and possibly ineffective, in sandstones.

Best fitting of induced-polarization (IP) spectra by different models of Cole-Cole type evidences discrepancies in the resulting model parameters. The time constant determined from the same data could vary in magnitude over several decades. This effect, which makes an evaluation of the results of different models nearly impossible, is demonstrated by induced polarization measurements in the frequency range between [Formula: see text] and [Formula: see text] on thirteen mixtures of quartz sand and slag grains. The samples differ in size and the amount of the slag grains. Parameters describing the IP spectra are derived by fitting models of the Cole-Cole type to the measured data. The fitting quality of the generalized Cole-Cole model, the standard Cole-Cole model, and the Cole-Davidson model is investigated. The parameters derived from these models are compared and correlated with mass percentage and grain size of the slag particles. An alternative fittingapproach is introduced, using the decomposition of observed IP spectra into a variety of Debye spectra. Four integrating parameters are derived and correlated with parameters of the slag-sand mixtures and Cole-Cole parameters, respectively. The alternative approach generally enables a better fitting of measured spectra compared with Cole-Cole type models. It proves to be more flexible and stable, even for complicated phase spectra that cannot be fitted by single Cole-Cole type models. The integrating parameters are well correlated with characterizing parameters of the slag-sand mixtures. The total chargeability well indicates the mass percentage of slag grains, and the mean relaxation time is related to the grain size. The relaxation time distribution can be displayed by cumulative normalized chargeability versus relaxation time, similar to granulation curves. Anologous to the latter, a nonuniformity parameter characterizes the width of the relaxation time distribution.