We describe an easily implementable method for non-destructive measurements of ultracold atomic clouds based on dark field imaging of spatially resolved Faraday rotation. The signal-to-noise ratio is analyzed theoretically and, in the absence of experimental imperfections, the sensitivity limit is found to be identical to other conventional dispersive imaging techniques. The dependence on laser detuning, atomic density, and temperature is characterized in a detailed comparison with theory. Due to low destructiveness, spatially resolved images of the same cloud can be acquired up to 2000 times. The technique is applied to avoid the effect of shot-to-shot fluctuations in atom number calibration, to demonstrate single-run vector magnetic field imaging and single-run spatial imaging of the system's dynamic behavior. This demonstrates that the method is a useful tool for the characterization of static and dynamically changing properties of ultracold atomic clouds.

We prepare number stabilized ultracold atom clouds through the real-time analysis of non-destructive images and the application of feedback. In our experiments, the atom number N ∼ 10 6 is determined by high precision Faraday imaging with uncertainty ∆N below the shot noise level, i.e., ∆N < √ N . Based on this measurement, feedback is applied to reduce the atom number to a user-defined target, whereupon a second imaging series probes the number stabilized cloud. By this method, we show that the atom number in ultracold clouds can be prepared below the shot noise level.Over the past decade, experiments with ultracold atomic samples have matured from the proof-of-concept level to a development platform for technologies such as quantum sensors and quantum simulators. One rapidly expanding technique is the manipulation of quantum systems using measurements and feedback [1][2][3]. To limit the back-action, usually a 'weak' measurement is employed, such as detecting the phase shift induced by an atomic ensemble on an off-resonant laser beam [4]. Recent experiments have demonstrated feedback control of motion in an optical lattice [5], a quantum memory for light [6], deterministic spin squeezing [7], stabilization of an atomic system against decoherence [8], extending the interrogation time in Ramsey experiments [9] and feedback cooling of a spin ensemble [10].To fully exploit the potential of ultracold clouds in emerging quantum technologies, these atomic samples must be prepared with unprecedented precision. For instance, precise atom number preparation is crucial to improve the precision of atomic clocks, which is presently limited by collisional shifts [11]. It is of particular relevance for techniques that employ interactions to produce non-classical states for improved interferometric sensitivity [12][13][14][15]. In general, if the number fluctuations of an atomic ensemble in a single spatial mode can be suppressed, the many-particle state becomes non-classical, yielding a resource for atom interferometry beyond the standard quantum limit [16].Sub-Poissonian preparation of micro-and mesoscopic atomic samples was recently demonstrated by using single-site addressing in an optical lattice [17], 3-body collisions [18,19], non-destructive measurements of nanofiber-based systems [20], and careful tailoring of the trapping potential for fermionic [21] and bosonic systems [22]. However, despite initial attempts towards the compensation of number fluctuations in ultracold atomic clouds [23], the high precision preparation of large atom numbers remains an unsolved challenge.In this Letter, we stabilize the atom number in ultracold clouds through the real-time analysis of dispersive images and feedback, as shown in Fig. 1(a). After initial evaporative cooling of an atomic cloud, a first set of non-destructive Faraday images, "F1", determines the number of atoms. We characterize this imaging method and show it achieves an atom number uncertainty below the shot noise level. Based on the analysis of the images, feedback is ...

The coherent manipulation of wave packets is an important tool in many areas of physics. We demonstrate the experimental realization of quasifree wave packets of ultra-cold atoms bound by an external harmonic trap. The wave packets are produced by modulating the intensity of an optical lattice containing a Bose-Einstein condensate. The evolution of these wave packets is monitored in situ and their six-photon reflection at a band gap is observed. In direct analogy with pump-probe spectroscopy, a probe pulse allows for the resonant de-excitation of the wave packet into states localized around selected lattice sites at a long, controllable distance of more than 100 lattice sites from the main component. This precise control mechanism for ultra-cold atoms thus enables controlled quantum state preparation and splitting for quantum dynamics, metrology and simulation.

We demonstrate that a dispersive imaging technique based on the Faraday effect can measure the atom number in a large, ultracold atom cloud with a precision below the atom shot noise level. The minimally destructive character of the technique allows us to take multiple images of the same cloud, which enables sub-atom shot noise measurement precision of the atom number and allows for an in situ determination of the measurement precision. We have developed a noise model that quantitatively describes the noise contributions due to photon shot noise in the detected light and the noise associated with single atom loss. This model contains no free parameters and is calculated through an analysis of the fluctuations in the acquired images. For clouds containing N ∼ 5 × 10 6 atoms, we achieve a precision more than a factor of two below the atom shot noise level.

We have investigated spin dynamics in a two-dimensional quantum gas. Through spin-changing collisions, two clouds with opposite spin orientations are spontaneously created in a Bose-Einstein condensate. After ballistic expansion, both clouds acquire ring-shaped density distributions with superimposed angular density modulations. The density distributions depend on the applied magnetic field and are well explained by a simple Bogoliubov model. We show that the two clouds are anticorrelated in momentum space. The observed momentum correlations pave the way towards the creation of an atom source with nonlocal Einstein-Podolsky-Rosen entanglement. Since the optical trapping of Bose-Einstein condensates (BECs) enabled the investigation of quantum gases with multiple spin components, spinor condensates have become a particularly rich research field [1,2]. While initial work focused on an understanding of the ground state and dynamical properties of spinor condensates [3][4][5], recent experiments have started to exploit their properties for applications in other fields. In particular, the production of entangled states through spin dynamics [6,7] has spawned interest in spin dynamics for their applications in precision metrology [8,9].Spin dynamics in a trapped quantum gas is strongly influenced by the geometry of the confining potential. In particular, highly asymmetric optical traps provide a way to reduce the dimensionality of a trapped quantum gas, both with respect to the motional and spin degrees of freedom [10]. Thus, tailored confining potentials offer new avenues for exploiting spin dynamics, e.g., the generation of correlated pairs of atoms in well-defined motional states [11], similar to work on four-wave mixing of ultracold atoms in an optical lattice [12][13][14].In this Rapid Communication, we investigate spin dynamics in a quantum gas confined to two dimensions (2D) by an optical lattice. We show how the spin excitation modes in the 2D potential lead to ring-shaped density distributions with a superimposed angular density modulation in time-of-flight images. The angular structure is traced to the matter-wave interference between multiple spin excitation modes with angular momentum. The observed density distributions may also be interpreted as several wave packets propagating in 2D with well-defined momentum.We investigate spin dynamics in a 87 Rb BEC prepared in |F = 2,m F = 0 (|0 ). By making several standard approximations to treat atomic collisions at ultralow temperatures, one finds that only collisions that preserve the total magnetization can occur [1]. Thus, the spin dynamics leads to scattering into |F = 2,m F = ±1 (|±1 ) and |F = 2,m F = ±2 (|±2 ), but for short evolution times, scattering between |0 and |±1 predominates; i.e., |0 + |0 ↔ |1 + |−1 . By treating the |0 condensate as a classical field ψ 0 and |±1 as small fluctuations δψ ±1 , the dynamics may be describedρ is the radial coordinate, ω ρ is the radial trapping frequency of the confining potential, and M is the mass. The 2D interacti...