Universal dynamics of rogue waves in a quenched spinor Bose condensate

Isolated many-body systems far from equilibrium may exhibit scaling dynamics with universal exponents indicating the proximity of the time-evolution to a non-thermal fixed point. We find universal dynamics connected with the occurrence of extreme wave excitations in the mutually coupled magnetic components of a spinor gas which propagate in an effectively random potential. The frequency of these rogue waves is affected by the time-varying spatial correlation length of the potential, giving rise to an additional exponent δ_c?1/3  for temporal scaling, which is different from the exponent β_V?1/4 characterizing the scaling of the correlation length l_V∼t^β_V in time. As a result of the caustics, i.e., focusing events, real-time instanton defects appear in the Larmor phase of the spin-1 system as vortices in space and time. The temporal correlations governing the instanton occurrence frequency scale as t^δ_I. This suggests that the universality class of a non-thermal fixed point could be characterized by different, mutually related exponents defining the evolution in time and space, respectively. Our results have a strong relevance for understanding pattern coarsening from first principles and potential implications for dynamics ranging from the early universe to geophysical dynamics and micro physics.

Data-driven discovery of statistically relevant information in quantum simulators

Quantum simulators offer powerful means to investigate strongly correlated quantum matter. However, interpreting measurement outcomes in such systems poses significant challenges. Here, we present a theoretical framework for information extraction in synthetic quantum matter, illustrated for the case of a quantum quench in a spinor Bose-Einstein condensate experiment. Employing non-parametric unsupervised learning tools that provide different measures of information content, we demonstrate a system-agnostic approach to identify dominant degrees of freedom. This enables us to rank operators according to their relevance, akin to effective field theory. To characterize the corresponding effective description, we then explore the intrinsic dimension of data sets as a measure of the complexity of the dynamics. This reveals a simplification of the data structure, which correlates with the emergence of time-dependent universal behavior in the studied system. Our assumption-free approach can be immediately applied in a variety of experimental platforms.

Universal Dynamics at the Lowest Temperatures

High-performance graphical processing units (GPU) are used for the repeated parallelised propagation of non-linear partial differential equations on large spatio-temporal grids. The main challenge results as a combination of the requirement of large grids for exploring scaling over several orders of magnitude, both in space and time, and the need for high statistics in averaging over many runs, in computing correlation functions for highly fluctuating quantum many-body states. With our simulations, we explore the dynamics of complex quantum systems far from equilibrium, with the aim of classifying their universal characteristics such as scaling exponents near non-thermal fixed points. Our results are strongly relevant for the development of synthetic quantum systems when exploring the respective physics in the laboratory.

Observation of two non-thermal fixed points for the same microscopic symmetry

Close to equilibrium, the underlying symmetries of a system determine its possible universal behavior. Far from equilibrium, however, different universal phenomena associated with the existence of multiple non-thermal fixed points can be realized for given microscopic symmetries. Here, we study this phenomenon using a quasi-one-dimensional spinor Bose-Einstein condensate. We prepare two different initial conditions and observe two distinct universal scaling dynamics with different exponents. Measurements of the complex-valued order parameter with spatial resolution allow us to characterize the phase-amplitude excitations for the two scenarios. Our study provides new insights into the phenomenon of universal dynamics far from equilibrium and opens a path towards mapping out the associated basins of non-thermal fixed points.

Universal Dynamics of Rogue Waves in a Quenched Spinor Bose Condensate

Isolated many-body systems far from equilibrium may exhibit scaling dynamics with universal exponents indicating the proximity of the time-evolution to a non-thermal fixed point. We find universal dynamics connected with the occurrence of extreme wave excitations in the mutually coupled magnetic components of a spinor gas which propagate in an effectively random potential. The frequency of these rogue waves is affected by the time-varying spatial correlation length of the potential, giving rise to an additional exponent δc~1/3 for temporal scaling, which is different from the exponent βV~1/4 characterizing the scaling of the correlation length λ?Vλ∼tβV in time. As a result of the caustics, i.e., focusing events, real-time instanton defects appear in the Larmor phase of the spin-1 system as vortices in space and time. The temporal correlations governing the instanton occurrence frequency scale as tδI. This suggests that the universality class of a non-thermal fixed point could be characterized by different, mutually related exponents defining the evolution in time and space, respectively. Our results have a strong relevance for understanding pattern coarsening from first principles and potential implications for dynamics ranging from the early universe to geophysical dynamics and micro physics.

Quantum approximate optimization algorithm for qudit systems

Condensation and thermalization of an easy-plane ferromagnet in a spinor Bose gas

Bose–Einstein condensates are an ideal platform to explore dynamical phenomena emerging in the many-body limit, such as the build-up of long-range coherence, superfluidity or spontaneous symmetry breaking. Here we study the thermalization dynamics of an easy-plane ferromagnet employing a homogeneous one-dimensional spinor Bose gas. We demonstrate the dynamic emergence of effective long-range coherence for the spin field and verify spin-superfluidity by experimentally testing Landau’s criterion. We reveal the structure of one massive and two massless emerging modes—a consequence of explicit and spontaneous symmetry breaking, respectively. Our experiments allow us to observe the thermalization of an easy-plane ferromagnetic Bose gas. The relevant momentum-resolved observables are in agreement with a thermal prediction obtained from an underlying microscopic model within the Bogoliubov approximation. Our methods and results are a step towards a quantitative understanding of condensation dynamics in large magnetic spin systems and the study of the role of entanglement and topological excitations for their thermalization.

Detecting Entanglement Structure in Continuous Many-Body Quantum Systems

A prerequisite for the comprehensive understanding of many-body quantum systems is a characterization in terms of their entanglement structure. The experimental detection of entanglement in spatially extended many-body systems describable by quantum fields still presents a major challenge. We develop a general scheme for certifying entanglement and demonstrate it by revealing entanglement between distinct subsystems of a spinor Bose-Einstein condensate. Our scheme builds on the spatially resolved simultaneous detection of the quantum field in two conjugate observables which allows the experimental confirmation of quantum correlations between local as well as nonlocal partitions of the system. The detection of squeezing in Bogoliubov modes in a multimode setting illustrates its potential to boost the capabilities of quantum simulations to study entanglement in spatially extended many-body systems.

Finding self-similar behavior in quantum many-body dynamics via persistent homology

Inspired by topological data analysis techniques, we introduce persistent homology observables and apply them in a geometric analysis of the dynamics of quantum field theories. As a prototype application, we consider data from a classical-statistical simulation of a two-dimensional Bose gas far from equilibrium. We discover a continuous spectrum of dynamical scaling exponents, which provides a refined classification of nonequilibrium self-similar phenomena. A possible explanation of the underlying processes is provided in terms of mixing strong wave turbulence and anomalous vortex kinetics components in point clouds. We find that the persistent homology scaling exponents are inherently linked to the geometry of the system, as the derivation of a packing relation reveals. The approach opens new ways of analyzing quantum many-body dynamics in terms of robust topological structures beyond standard field theoretic techniques.

Analog cosmological reheating in an ultracold Bose gas

Cosmological reheating describes the transition of the postinflationary universe to a hot and thermal state. In order to shed light on the underlying dynamics of this process, we propose to quantum-simulate the reheating-like dynamics of a generic cosmological single-field model in an ultracold Bose gas. In our setup, the excitations on top of an atomic Bose-Einstein condensate play the role of the particles produced by the decaying inflaton field after inflation. Expanding spacetime as well as the background oscillating inflaton field are mimicked in the nonrelativistic limit by a time dependence of the atomic interactions, which can be tuned experimentally via Feshbach resonances. As we illustrate by means of classical-statistical simulations for the case of two spatial dimensions, the dynamics of the atomic system exhibits the characteristic stages of far-from-equilibrium reheating, including the amplification of fluctuations via parametric instabilities and the subsequent turbulent transport of energy towards higher momenta. The transport is governed by a nonthermal fixed point showing universal self-similar time evolution as well as a transient regime of prescaling with time-dependent scaling exponents. While the classical-statistical simulations can capture only the earlier stages of the dynamics for weak couplings, the proposed experiment has the potential of exploring the evolution up to late times even beyond the weak coupling regime.

Universell in der Zeit

https://www.pro-physik.de/physik-journal/februar-2021

Collisions of Three-Component Vector Solitons in Bose-Einstein Condensates

Ultracold gases provide an unprecedented level of control for the investigation of soliton dynamics and collisions. We present a scheme for deterministically preparing pairs of three-component solitons in a Bose-Einstein condensate. Our method is based on local spin rotations which simultaneously imprint suitable phase and density distributions. This enables us to observe striking collisional properties of the vector degree of freedom which naturally arises for the coherent nature of the emerging multicomponent solitons. We find that the solitonic properties in the quasi-one-dimensional system are quantitatively described by the integrable repulsive three-component Manakov model.

Spiking neuromorphic chip learns entangled quantum states

The approximation of quantum states with artificial neural networks has gained a lot of attention during the last years. Meanwhile, analog neuromorphic chips, inspired by structural and dynamical properties of the biological brain, show a high energy efficiency in running artificial neural-network architectures for the profit of generative applications. This encourages employing such hardware systems as platforms for simulations of quantum systems. Here we report on the realization of a prototype using the latest spike-based BrainScaleS hardware allowing us to represent few-qubit maximally entangled quantum states with high fidelities. Bell correlations of pure and mixed two-qubit states are well captured by the analog hardware, demonstrating an important building block for simulating quantum systems with spiking neuromorphic chips.

Wie Quantensimulatoren neue Antworten geben können

https://www.pro-physik.de/physik-journal/februar-2021

Experimental extraction of the quantum effective action for a non-equilibrium many-body system

Stability analysis of ground states in a one-dimensional trapped spin-1 Bose gas

In this work we study the stability properties of the ground states of a spin-1 Bose gas in presence of a trapping potential in one spatial dimension. To set the stage we first map out the phase diagram for the trapped system by making use of a, so-called, continuous-time Nesterov method. We present an extension of the method, which has been previously applied to one-component systems, to our multi-component system. We show that it is a powerful and robust tool for finding the ground states of a physical system without the need of an accurate initial guess. We subsequently solve numerically the Bogoliubov de-Gennes equations in order to analyze the stability of the ground states of the trapped spin-1 system. We find that the trapping potential retains the overall structure of the stability diagram, while affecting the spectral details of each of the possible ground state waveforms. It is also found that the peak density of the trapped system is the characteristic quantity describing dynamical instabilities in the system. Therefore replacing the homogeneous density with the peak density of the trapped system leads to good agreement of the homogeneous Bogoliubov predictions with the numerically observed maximal growth rates of dynamically unstable modes. The stability conclusions in the one-dimensional trapped system are independent of the spin coupling strength and the normalized trap strength over several orders of magnitude of their variation.

Simultaneous Readout of Noncommuting Collective Spin Observables beyond the Standard Quantum Limit

Bidirectional universal dynamics in a spinor Bose gas close to a nonthermal fixed point

We numerically study the universal scaling dynamics of an isolated one-dimensional ferromagnetic spin-1 Bose gas. Preparing the system in a far-from-equilibrium initial state, simultaneous coarsening and refining is found to enable and characterize the approach to a nonthermal fixed point. A macroscopic length scale which scales in time according to $L_{\Lambda }\left(t\right)\sim t^{\beta }$, with $\beta ?1/4$, quantifies the coarsening of the size of spin textures. At the same time kinklike defects populating these textures undergo a refining process measured by a shrinking microscopic length scale $L_{\lambda }\sim t^{{\beta }^{\prime }}$, with ${\beta }^{\prime }?-0.17$. The combination of these scaling evolutions enables particle and energy conservation in the isolated system and constitutes a bidirectional transport in momentum space. The value of the coarsening exponent $\beta$ suggests the dynamics to belong to the universality class of diffusive coarsening of the one-dimensional $XY$ model. However, the universal momentum distribution function exhibiting nonlinear transport marks the distinction between diffusive coarsening and the approach of a nonthermal fixed point in the isolated system considered here. This underlines the importance of the universal scaling function in classifying nonthermal fixed points. Present-day experiments with quantum gases are expected to have access to the predicted bidirectional scaling.

Observation of universal dynamics in a spinor Bose gas far from equilibrium

The dynamics of quantum systems far from equilibrium represents one of the most challenging problems in theoretical many-body physics. While the evolution is in general intractable in all its details, relevant observables can become insensitive to microscopic system parameters and initial conditions. This is the basis of the phenomenon of universality. Far from equilibrium, universality is identified through the scaling of the spatio-temporal evolution of the system, captured by universal exponents and functions. Theoretically, this has been studied in examples as different as the reheating process in inflationary universe cosmology, the dynamics of nuclear collision experiments described by quantum chromodynamics, or the post-quench dynamics in dilute quantum gases in non-relativistic quantum field theory. However, an experimental demonstration of such scaling evolution in space and time in a quantum many-body system is lacking so far. Here we observe the emergence of universal dynamics by evaluating spatially resolved spin correlations in a quasi one-dimensional spinor Bose-Einstein condensate. For long evolution times we extract the scaling properties from the spatial correlations of the spin excitations. From this we find the dynamics to be governed by transport of an emergent conserved quantity towards low momentum scales. Our results establish an important class of non-stationary systems whose dynamics is encoded in time-independent scaling exponents and functions signaling the existence of non-thermal fixed points. We confirm that the non-thermal scaling phenomenon involves no fine-tuning, by preparing different initial conditions and observing the same scaling behaviour. Our analog quantum simulation approach provides the basis to reveal the underlying mechanisms and characteristics of non-thermal universality classes. One may use this universality to learn, from experiments with ultra-cold gases, about fundamental aspects of dynamics studied in cosmology and quantum chromodynamics.

Spatially distributed multipartite entanglement enables EPR steering of atomic clouds

A key resource for distributed quantum-enhanced protocols is entanglement between spatially separated modes. However, the robust generation and detection of entanglement between spatially separated regions of an ultracold atomic system remain a challenge. We used spin mixing in a tightly confined Bose-Einstein condensate to generate an entangled state of indistinguishable particles in a single spatial mode. We show experimentally that this entanglement can be spatially distributed by self-similar expansion of the atomic cloud. We used spatially resolved spin read-out to reveal a particularly strong form of quantum correlations known as Einstein-Podolsky-Rosen (EPR) steering between distinct parts of the expanded cloud. Based on the strength of EPR steering, we constructed a witness, which confirmed genuine 5-partite entanglement.

Low-energy effective theory of non-thermal fixed points in a multicomponent Bose gas

Non-thermal fixed points in the evolution of a quantum many-body system quenched far out of equilibrium manifest themselves in a scaling evolution of correlations in space and time. We develop a low-energy effective theory of non-thermal fixed points in a bosonic quantum many-body system by integrating out long-wave-length density fluctuations. The system consists of N distinguishable spatially uniform Bose gases with O(N)×U(1)-symmetric interactions. The effective theory describes interacting Goldstone modes of the total and relative-phase excitations. It is similar in character to the non-linear Luttinger-liquid description of low-energy phonons in a single dilute Bose gas, with the markable difference of a universal non-local coupling function depending, in the large-N limit, only on momentum, single-particle mass, and density of the gas. Our theory provides a perturbative description of the non-thermal fixed point, technically easy to apply to experimentally relevant cases with a small number of fields N. Numerical results for N=3 allow us to characterize the analytical form of the scaling function and confirm the analytically predicted scaling exponents. The fixed point which is dominated by the relative phases is found to be Gaussian, while a non-Gaussian fixed point is anticipated to require scaling evolution with a distinctly lower power of time.

Prescaling in a far-from-equilibrium Bose gas

Non-equilibrium conditions give rise to a class of universally evolving low-energy configurations of fluctuating dilute Bose gases at a non-thermal fixed point. While the fixed point and thus full scaling in space and time is generically only reached at very long evolution times, we here propose that systems can show prescaling much earlier, on experimentally accessible time scales. During the prescaling evolution, some well-measurable short-distance properties of the spatial correlations already scale with the universal exponents of the fixed point while others still show scaling violations. Prescaling is characterized by the evolution obeying already, to a good approximation, the conservation laws which are associated with the asymptotically reached non-thermal fixed point, defining its belonging to a specific universality class. In our simulations, we consider N=3 spatially uniform three-dimensional Bose gases of particles labeled, e.g., by different hyperfine magnetic quantum numbers, with identical inter- and intra-species interactions. In this system, the approach of a non-thermal fixed point is marked by low-energy phase excitations self-similarly redistributing towards smaller wave numbers. During prescaling, the full U(N) symmetry of the model is broken while the conserved transport, reflecting the remaining U(1) symmetries, leads to the buildup of a rescaling quasicondensate distribution.

Quenches near criticality of the quantum Ising chain-power and limitations of the discrete truncated Wigner approximation

The semi-classical discrete truncated Wigner approximation (dTWA) has recently been proposed as a simulation method for spin-1/2 systems. While it appears to provide a powerful approach which shows promising results in higher dimensions and for systems with long-range interactions, its performance is still not well understood in general. Here we perform a systematic benchmark on the one-dimensional transverse-field Ising model and point to limitations of the approximation arising after sudden quenches into the quantum critical regime. Our procedure allows to identify the limitations of the semi-classical simulations and with that to determine the regimes and questions where quantum simulators can provide information which is inaccessible to semi-classics.

Active SU(1,1) atom interferometry

Active interferometers use amplifying elements for beam splitting and recombination. We experimentally implement such a device by using spin exchange in a Bose–Einstein condensate. The two interferometry modes are initially empty spin states that get spontaneously populated in the process of parametric amplification. This nonlinear mechanism scatters atoms into both modes in a pairwise fashion and generates a non-classical state. Finally, a matched second period of spin exchange is performed that nonlinearly amplifies the output signal and maps the phase onto readily detectable first moments. Depending on the accumulated phase this nonlinear readout can reverse the initial dynamics and deamplify the entangled state back to empty spin states. This sequence is described in the framework of SU(1,1) mode transformations and compared to the SU(2) angular momentum description of passive interferometers.

Universal equilibrium scaling functions at short times after a quench

By analyzing spin-spin correlation functions at relatively short distances, we show that equilibrium near-critical properties can be extracted at short times after quenches into the vicinity of a quantum critical point. The time scales after which equilibrium properties can be extracted are sufficiently short so that the proposed scheme should be viable for quantum simulators of spin models based on ultracold atoms or trapped ions. Our results, analytic as well as numeric, are for one-dimensional spin models, either integrable or nonintegrable, but we expect our conclusions to be valid in higher dimensions as well.

Implementing quantum electrodynamics with ultracold atomic systems

We discuss the experimental engineering of model systems for the description of QED in one spatial dimension via a mixture of bosonic ²³Na and fermionic ⁶Li atoms. The local gauge symmetry is realized in an optical superlattice, using heteronuclear boson-fermion spin-changing interactions which preserve the total spin in every local collision. We consider a large number of bosons residing in the coherent state of a Bose-Einstein condensate on each link between the fermion lattice sites, such that the behavior of lattice QED in the continuum limit can be recovered. The discussion about the range of possible experimental parameters builds, in particular, upon experiences with related setups of fermions interacting with coherent samples of bosonic atoms. We determine the atomic system's parameters required for the description of fundamental QED processes, such as Schwinger pair production and string breaking. This is achieved by benchmark calculations of the atomic system and of QED itself using functional integral techniques. Our results demonstrate that the dynamics of one-dimensional QED may be realized with ultracold atoms using state-of-the-art experimental resources. The experimental setup proposed may provide a unique access to longstanding open questions for which classical computational methods are no longer applicable.

Quantum-Enhanced Sensing Based on Time Reversal of Nonlinear Dynamics

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 quantum-enhanced 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 dressed states at the miscibility-immiscibility threshold

The dynamical evolution of spatial patterns in a complex system can reveal the underlying structure and stability of stationary states. As a model system we employ a two-component Bose-Einstein condensate at the transition from miscible to immiscible with the additional control of linear interconversion. Excellent agreement is found between the detailed experimental time evolution and the corresponding numerical mean-field computations. Analyzing the dynamics of the system, we find clear indications of stationary states that we term nonlinear dressed states. A steady-state bifurcation analysis reveals a smooth connection of these states with dark-bright soliton solutions of the integrable two-component Manakov model.

Observation of Scaling in the Dynamics of a Strongly Quenched Quantum Gas

We report on the experimental observation of scaling in the time evolution following a sudden quench into the vicinity of a quantum critical point. The experimental system, a two-component Bose gas with coherent exchange between the constituents, allows for the necessary high level of control of parameters as well as the access to time-resolved spatial correlation functions. The theoretical analysis reveals that when quenching the system close to the critical point, the energy introduced by the quench leads to a short-time evolution exhibiting crossover reminiscent of the finite-temperature critical properties in the system’s universality class. Observing the time evolution after a quench represents a paradigm shift in accessing and probing experimentally universal properties close to a quantum critical point and allows in a new way benchmarking of quantum many-body theory with experiments.

Twist-and-turn spin squeezing in Bose-Einstein condensates

We demonstrate experimentally an alternative method for the dynamic generation of atomic spin squeezing, building on the interplay between linear coupling and nonlinear phase evolution. Since the resulting quantum dynamics can be seen as rotation and shear on the generalized Bloch sphere, we call this scheme twist-and-turn. This is closely connected to an underlying instability in the classical limit of this system. The short-time evolution of the quantum state is governed by a fast initial spreading of the quantum uncertainty in one direction, accompanied by squeezing in the orthogonal axis. We find an optimal value of ξ²_S=−7.1(3) dB in a single Bose-Einstein condensate and scalability of the squeezing to more than 10⁴ particles with ξ²_S=−2.8(4) dB.

Scalable Spin Squeezing for Quantum-Enhanced Magnetometry with Bose-Einstein Condensates

A major challenge in quantum metrology is the generation of entangled states with a macroscopic atom number. Here, we demonstrate experimentally that atomic squeezing generated via nonlinear dynamics in Bose-Einstein condensates, combined with suitable trap geometries, allows scaling to large ensemble sizes. We achieve a suppression of fluctuations by 5.3(5) dB for 12 300 particles, from which we infer that similar squeezing can be obtained for more than 10⁷ atom. With this resource, we demonstrate quantum-enhanced magnetometry by swapping the squeezed state to magnetically sensitive hyperfine levels that have negligible nonlinearity. We find a quantum-enhanced single-shot sensitivity of 310(47) pT for static magnetic fields in a probe volume as small as 90 μm³.

Fisher information and entanglement of non-Gaussian spin states

Entanglement is the key quantum resource for improving measurement sensitivity beyond classical limits. However, the production of entanglement in mesoscopic atomic systems has been limited to squeezed states, described by Gaussian statistics. Here, we report on the creation and characterization of non-Gaussian many-body entangled states. We develop a general method to extract the Fisher information, which reveals that the quantum dynamics of a classically unstable system creates quantum states that are not spin squeezed but nevertheless entangled. The extracted Fisher information quantifies metrologically useful entanglement, which we confirm by Bayesian phase estimation with sub–shot-noise sensitivity. These methods are scalable to large particle numbers and applicable directly to other quantum systems.

Dissipative preparation of phase- and number-squeezed states with ultracold atoms

We develop a dissipative quantum state preparation scheme for the creation of phase- and number-squeezed states. It utilizes ultracold atoms in a double-well configuration immersed in a background Bose-Einstein condensate, with the latter consisting of an atom species different from the atoms in the double well and acting as a dissipative quantum reservoir. We derive a master equation for this system starting from microscopic physics and show that squeezing develops on a time scale proportional to 1/N, where N is the number of particles in the double well. This scaling, caused by bosonic enhancement, allows us to make the time scale for the creation of squeezed states very short. The lifetime of squeezed states is limited by dephasing arising from the intrinsic structure of the setup. However, the dephasing can be avoided by stroboscopically switching the driving off and on. We show that this approach leads to robust stationary squeezed states. Finally, we provide the necessary ingredients for a potential experimental implementation by specifying a parameter regime for rubidium atoms that leads to squeezed states.

Accurate Atom Counting in Mesoscopic Ensembles

Many cold atom experiments rely on precise atom number detection, especially in the context of quantum-enhanced metrology where effects at the single particle level are important. Here, we investigate the limits of atom number counting via resonant fluorescence detection for mesoscopic samples of trapped atoms. We characterize the precision of these fluorescence measurements beginning from the single-atom level up to more than one thousand. By investigating the primary noise sources, we obtain single-atom resolution for atom numbers as high as 1200. This capability is an essential prerequisite for future experiments with highly entangled states of mesoscopic atomic ensembles.

Optimized absorption imaging of mesoscopic atomic clouds

We report on the optimization of high-intensity absorption imaging for small Bose–Einstein condensates. The imaging calibration exploits the linear scaling of the quantum projection noise with the mean number of atoms for a coherent spin state. After optimization for atomic clouds containing up to 300 atoms, we find an atom number resolution of Δ det =3.7 Δ det = 3.7 atoms, mainly limited by photon shot noise and radiation pressure.

Measuring and engineering entropy and spin squeezing in weakly linked Bose-Einstein condensates

We propose a method to infer the single-particle entropy of bosonic atoms in an optical lattice and to study the local evolution of entropy, spin squeezing, and entropic inequalities for entangle- ment detection in such systems. This method is based on experimentally feasible measurements of non-nearest-neighbour coherences. We study a specific example of dynamically controlling atom tunneling between selected sites and show that this could potentially also improve the metrologically relevant spin squeezing.

Dynamic generation of spin-squeezed states in bosonic Josephson junctions

We analyze the formation of squeezed states in a condensate of ultracold bosonic atoms confined by a double-well potential. The emphasis is set on the dynamical formation of such states from initially coherent many-body quantum states. Two cases are described: the squeezing formation in the evolution of the system around the stable point, and in the short-time evolution in the vicinity of an unstable point. The latter is shown to produce highly squeezed states on very short times. On the basis of a semiclassical approximation to the Bose-Hubbard Hamiltonian, we are able to predict the amount of squeezing, its scaling with N, and the speed of coherent spin formation with simple analytical formulas which successfully describe the numerical Bose-Hubbard results. This method of producing highly squeezed spin states in systems of ultracold atoms is compared to other standard methods in the literature.

Atomic homodyne detection of continuous variable entangled twin-atom states

Historically, the completeness of quantum theory has been questioned using the concept of bipartite continuous-variable entanglement. The non-classical correlations (entanglement) between the two subsystems imply that the observables of one subsystem are determined by the measurement choice on the other, regardless of the distance between the subsystems. Nowadays, continuous-variable entanglement is regarded as an essential resource, allowing for quantum enhanced measurement resolution, the realization of quantum teleportation and quantum memories or the demonstration of the Einstein–Podolsky–Rosen paradox. These applications rely on techniques to manipulate and detect coherences of quantum fields, the quadratures. Whereas in optics coherent homodyne detection of quadratures is a standard technique, for massive particles a corresponding method was missing. Here we report the realization of an atomic analogue to homodyne detection for the measurement of matter-wave quadratures. The application of this technique to a quantum state produced by spin-changing collisions in a Bose–Einstein condensate reveals continuous-variable entanglement, as well as the twin-atom character of the state. Our results provide a rare example of continuous-variable entanglement of massive particles. The direct detection of atomic quadratures has applications not only in experimental quantum atom optics, but also for the measurement of fields in many-body systems of massive particles.

Rabi Flopping Induces Spatial Demixing Dynamics

We experimentally investigate the mixing and demixing dynamics of Bose-Einstein condensates in the presence of a linear coupling between two internal states. The observed amplitude reduction of the Rabi oscillations can be understood as a result of demixing dynamics of dressed states as experimentally confirmed by reconstructing the spatial profile of dressed state amplitudes. The observations are in quantitative agreement with numerical integration of coupled Gross-Pitaevskii equations without free parameters, which also reveals the criticality of the dynamics on the symmetry of the system. Our observations demonstrate new possibilities for changing effective atomic interactions and studying critical phenomena.

Local and spatially extended sub-Poisson atom-number fluctuations in optical lattices

We demonstrate that ultracold interacting bosonic atoms in an optical lattice with large on-site population show sub-Poisson on-site and intersite atom-number fluctuations. The experimental observations agree with  numerical predictions of the truncated Wigner approximation. The correlations persist in the presence of multimode atom dynamics and even over large spatially extended samples involving several sites.

Two-Well Bose-Einstein Condensates and Strategies for Einstein-Podolsky-Rosen Entanglement

Criteria suitable for measuring entanglement between two different potential wells in a Bose-Einstein condensation are evaluated. We show how to generate the required entanglement, utilizing either an adiabatic two-mode or a dynamic four-mode interaction strategy, with techniques that take advantage of s-wave scattering interactions to provide the nonlinear coupling. The dynamic entanglement method results in an entanglement signature with spatially separated detectors, as in the Einstein-Podolsky-Rosen paradox.

Einstein-Podolsky-Rosen Correlations of Ultracold Atomic Gases

We demonstrate that collective continuous variables of two species of trapped ultracold bosonic gases can be Einstein-Podolsky-Rosen-correlated (entangled) via inherent interactions between the species. We propose two different schemes for creating these correlations—a dynamical scheme and a static scheme analogous to two-mode squeezing in quantum optics. We quantify the correlations by using known measures of entanglement and study the effect of finite temperature on these quantum correlations.

Classical Bifurcation at the Transition from Rabi to Josephson Dynamics

We report on the experimental demonstration of the internal bosonic Josephson effect in a rubidium spinor Bose-Einstein condensate. The measurement of the full time dynamics in phase space allows the characterization of the theoretically predicted π-phase modes and quantitatively confirms analytical predictions, revealing a classical bifurcation. Our results suggest that this system is a model system which can be tuned from classical to the quantum regime and thus is an important step towards the experimental investigation of entanglement generation close to critical points.

Multiple atomic dark solitons in cigar-shaped Bose-Einstein condensates

We consider the stability and dynamics of multiple dark solitons in cigar-shaped Bose-Einstein condensates. 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 multisoliton states emerge as a nonlinear continuation of the appropriate excited eigenstates 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 are 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 multisoliton states including a three-soliton state consisting of two solitons oscillating around a stationary one and compare the relevant results to the predictions of the theoretical mean-field model.

Two-mode Bose gas: Beyond classical squeezing

The dynamical evolution of squeezing correlations in an ultracold Bose-Einstein condensate distributed across two modes is investigated theoretically in the framework of the Bose-Hubbard model. It is shown that the eigenstates of the Hamiltonian do not exploit the full region allowed by Heisenberg’s uncertainty relation for number and phase fluctuations. The development of nonclassical correlations and relative number squeezing is studied in the transition from the Josephson to the Fock regime. Comparing the full quantum evolution with classical statistical simulations allows us to identify quantum aspects of the squeezing formation. In the quantum regime, the measurement of squeezing allows us to distinguish even and odd total particle numbers.

Nonlinear atom interferometer surpasses classical precision limit

Interference is fundamental to wave dynamics and quantum mechanics. The quantum wave properties of particles are exploited in metrology using atom interferometers, allowing for high-precision inertia measurements^1,2. Furthermore, the state-of-the-art time standard is based on an interferometric technique known as Ramsey spectroscopy. However, the precision of an interferometer is limited by classical statistics owing to the finite number of atoms used to deduce the quantity of interest³. Here we show experimentally that the classical precision limit can be surpassed using nonlinear atom interferometry with a Bose–Einstein condensate. Controlled interactions between the atoms lead to non-classical entangled states within the interferometer; this represents an alternative approach to the use of non-classical input states^{4, 5, 6, 7, 8}. Extending quantum interferometry⁹ to the regime of large atom number, we find that phase sensitivity is enhanced by 15 per cent relative to that in an ideal classical measurement. Our nonlinear atomic beam splitter follows the ‘one-axis-twisting’ scheme¹⁰ and implements interaction control using a narrow Feshbach resonance. We perform noise tomography of the quantum state within the interferometer and detect coherent spin squeezing with a squeezing factor of -8.2 dB (refs 11–15). The results provide information on the many-particle quantum state, and imply the entanglement of 170 atoms¹⁶.

Ultrakalte Quantenpendel

Das Phänomen des Tunnelns – das Durchdringen von Barrieren – ist ein Paradebeispiel dafür, dass sich Teilchen in der Quantenwelt anders verhalten als in der klassischen. Kürzlich gelang es erstmals, das Tunneln makroskopischer Atomgase im Experiment direkt zu beobachten. Unter gewissen Näherungen lässt sich das komplizierte Vielteilchensystem mit einem einzigen Freiheitsgrad beschreiben, dessen Bewegungsgleichung derjenigen des Fadenpendels entspricht.

Dynamic control and probing of many-body decoherence in double-well Bose-Einstein condensates

Feynman path-integral treatment of the BEC-impurity polaron

Experimental Observation of Oscillating and Interacting Matter Wave Dark Solitons

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). The oscillation of the solitons is observed and the frequency is in quantitative agreement with simulations using the Gross-Pitaevskii equation. An effective particle picture is developed and reveals that the deviation of the observed frequencies from the asymptotic prediction ν_z/√2, where ν_z is the longitudinal trapping frequency, results from the dimensionality of the system and the interaction between the solitons.

Effective parameters for weakly coupled Bose-Einstein condensates

The dynamics of quantum fluctuations of weakly coupled Bose–Einstein condensates (BECs) can be described by an effective Bose–Josephson Hamiltonian. By requiring that the mean-field approximation on this effective Hamiltonian reproduces the low energy dynamics of the Gross–Pitaevskii equation, we obtain parameters for the effective Hamiltonian. This approach is particularly suitable when the BECs are in the Thomas–Fermi regime. Considering the problem of the splitting of a trapped BEC into two BEC fragments, our results for the dynamics of the depletion, collapses and revivals of the phase coherence are in good agreement with a recent numerical microscopic calculation from Streltsov et al (2007 Phys. Rev. Lett. 99 030402). In addition, the excitation energy of the lowest symmetric mode, which is the first relevant mode for the symmetric splitting process, is reproduced with reasonable accuracy all the way from the mean-field Josephson regime to the Fock regime.

Squeezing and entanglement in a Bose-Einstein condensate

Entanglement, a key feature of quantum mechanics, is a resource that allows the improvement of precision measurements beyond the conventional bound attainable by classical means. This results in the standard quantum limit, which is reached in today"s best available sensors of various quantities such as time and position. Many of these sensors are interferometers in which the standard quantum limit can be overcome by using quantum-entangled states (in particular spin squeezed states) at the two input ports. Bose–Einstein condensates of ultracold atoms are considered good candidates to provide such states involving a large number of particles. Here we demonstrate spin squeezed states suitable for atomic interferometry by splitting a condensate into a few parts using a lattice potential. Site-resolved detection of the atoms allows the measurement of the atom number difference and relative phase, which are conjugate variables. The observed fluctuations imply entanglement between the particles, a resource that would allow a precision gain of 3.8 dB over the standard quantum limit for interferometric measurements.

Solitonen im Bose-Einstein-Kondensat

Bose-Einstein-Kondensate sind perfekte Quantensysteme für das Studium der Dynamik von Wellenpaketen. Ein mächtiges Werkzeug sind dabei periodische optische Potentiale aus Laserlicht. Ein solches optisches Gitter wirkt auf das Wellenpaket aus ultrakalten Atomen wie ein atomares Kristallgitter auf freie Elektronen. Dadurch entstehen Energiebänder. Weil die Potentialtiefe des optischen Gitters frei einstellbar ist, wird so die effektive Masse der Wellenpakete elegant steuerbar. Sogar eine negative effektive Masse ist im optischen Gitter realisierbar. Mit ihr lässt sich das Zerfließen eines Wellenpakets zeitlich umkehren. Das System ermöglicht auch die Erzeugung heller Solitonen, also nichtzerfließender Wellenpakete, obwohl sich die Atome untereinander abstoßen. Bislang wurden diese aus Wolken mit etwa 350 Rubidium-Atomen erzeugt.

A primary noise thermometer for ultracold bose gases

We discuss in detail the experimental investigation of thermally induced fluctuations of the relative phase between two weakly coupled Bose–Einstein condensates (BECs). In analogy to superconducting Josephson junctions, the weak coupling originates from a tunnelling process through a potential barrier which is obtained by trapping the condensates in an optical double-well potential. The observed fluctuations of the relative phase are in quantitative agreement with a many body two mode model at finite temperature. The agreement demonstrates the possibility of using the phase fluctuation measurements in a bosonic Josephson junction (BJJ) as a primary thermometer. This new method allows for measuring temperatures far below the critical temperature where standard methods based on time of flight measurements fail. We employ this new thermometer to probe the heat capacity of a degenerate Bose gas as a function of temperature.

Noise Thermometry with Two Weakly Coupled Bose-Einstein Condensates

Here we report on the experimental investigation of thermally induced fluctuations of the relative phase between two Bose-Einstein condensates which are coupled via tunneling. The experimental control over the coupling strength and the temperature of the thermal background allows for the quantitative analysis of the phase fluctuations. Furthermore, we demonstrate the application of these measurements for thermometry in a regime where standard methods fail. With this we confirm that the heat capacity of an ideal Bose gas deviates from that of a classical gas as predicted by the third law of thermodynamics.

Realization of a single Josephson junction for Bose-Einstein condensates

We report on the realization of a double-well potential for Rubidium-87 Bose-Einstein condensates. The experimental setup allows for the investigation of two different dynamical phenomena known for this system – Josephson oscillations and self-trapping. We give a detailed discussion of the experimental setup and the methods used for calibrating the relevant parameters. We compare our experimental findings with the predictions of an extended two-mode model and find quantitative agreement.

Direct Observation of Tunneling and Nonlinear Self-Trapping in a single Bosonic Weak Link

We report on the first realization of a single bosonic Josephson junction, implemented by two weakly linked Bose-Einstein condensates in a double-well potential. In order to fully investigate the nonlinear tunneling dynamics we measure the density distribution in situ and deduce the relative phase between the two condensates from interference fringes. Our results verify the predicted nonlinear generalization of tunneling oscillations in superconducting and superfluid Josephson junctions. Additionally we confirm a novel nonlinear effect known as macroscopic quantum self-trapping, which leads to the inhibition of large amplitude tunneling oscillations.

Nonlinear Self-Trapping of Matter Waves in Periodic Potentials

We report the first experimental observation of nonlinear self-trapping of Bose-condensed 87Rb atoms in a one dimensional waveguide with a superimposed deep periodic potential . The trapping effect is confirmed directly by imaging the atomic spatial distribution. Increasing the nonlinearity we move the system from the diffusive regime, characterized by an expansion of the condensate, to the nonlinearity dominated self-trapping regime, where the initial expansion stops and the width remains finite. The data are in quantitative agreement with the solutions of the corresponding discrete nonlinear equation. Our results reveal that the effect of nonlinear self-trapping is of local nature, and is closely related to the macroscopic self-trapping phenomenon already predicted for double-well systems.

Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction

Bright Bose-Einstein Gap Solitons of Atoms with Repulsive Interaction

We report on the first experimental observation of bright matter wave solitons for 87Rb atoms with repulsive atom-atom interaction. This counterintuitive situation arises inside a weak periodic potential, where anomalous dispersion can be realized at the Brillouin zone boundary. If the coherent atomic wave packet is prepared at the corresponding band edge, a bright soliton is formed inside the gap. The strength of our system is the precise control of preparation and real time manipulation, allowing the systematic investigation of gap solitons

Linear and nonlinear dynamics of matter wave packets in periodic potentials

Nonlinear Self-Trapping of Matter Waves in Periodic Potentials