Moritz Hiller, Hannah Venzl, Tobias Zech, Bartlomiej Oleś, Florian Mintert, Andreas Buchleitner
We identify regular structures in the globally chaotic spectra of an interacting bosonic quantum gas in tilted periodic potentials. The associated eigenstates exhibit strong localization properties on the lattice, and are dynamically robust against external perturbations.
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We discuss the dynamical response of strongly interacting Bose atoms in an adiabatically tilted optical lattice. The analysis is performed in terms of the multi-level Landau-Zenner tunneling. Different regimes of tunneling are identified and analytical expressions for the doublon number, which is the quantity measured in laboratory experiments, are derived.
We present a sequence of driven-dissipative protocols for controlling cold atoms in tilted optical lattices. These experimentally accessible examples are templates that demonstrate how dissipation can be used to manipulate quantum many-body systems. We consider bosonic atoms trapped in a tilted optical lattice, immersed in a superfluid bath, and excited by coherent Raman lasers. With these ingredients, we are able to controllably transport atoms in the lattice and produce self-healing quantum states: a Mott insulator and the topologically ordered spin-1 AKLT state.
We suggest a method for creating bright matter solitons by loading a BEC of atoms in a driven tilted optical lattice. It is shown that one can realize the self-focussing regime for the wave-packet dynamics by properly adjusting the phase of the driving field with respect to the phase of Bloch oscillations. If atom-atom interactions are larger than some critical value $g_{min}$, this self-focussing regime is followed by the formation of bright solitons. Increasing the interactions above another critical value $g_{max}$ makes this process unstable. Instead of soliton formation one now meets the …
We analyze the energy spectrum and eigenstates of cold atoms in a tilted brick-wall optical lattice. When the tilt is applied, the system exhibits a sequence of topological phase transitions reflected in an abrupt change of the eigenstates. It is demonstrated that these topological phase transitions can be easily detected in a laboratory experiment by observing Bloch oscillations of cold atoms.
The dynamics of cold Bose atoms in driven tilted optical lattices is analyzed focusing on destruction of Wannier-Stark localization and the phenomenon of band collapse. It is argued that an understanding of the experimental results requires thorough account for interaction effects. These are suppression of the ballistic spreading of atoms for resonant driving (a multiple of the driving frequency coincides with the Bloch frequency) and unbounded sub-diffusive spreading of atoms for off-resonant driving.
Characterization of the chaotic phase in the tilted Bose-Hubbard model
研究通过能量、倾斜强度和粒子相互作用分析倾斜玻色-哈伯德模型的混沌相，发现适度倾斜可增强原始哈伯德哈密顿量的混沌相，并从均匀密度配置角度揭示混沌区域的发展与标度，为未来研究提供相图。
We present a brief overview of the phases and dynamics of ultracold bosons in an optical lattice in the presence of a tilt. We begin with a brief summary of the possible experimental setup for generating the tilt. This is followed by a discussion of the effective low-energy model for these systems and its equilibrium phases. We also chart the relation of this model to the recently studied system of ultracold Rydberg atoms. Next, we discuss the non-equilibrium dynamics of this model for quench, ramp and periodic protocols with emphasis on the periodic drive which can be understood in terms of a…
We discuss the quantum phase transition from the Mott-insulator state to the density-wave state for cold Bose atoms in a 2D square lattice as the lattice is adiabatically tilted along one of its primary axes. It is shown that a small misalignment of the tilt drastically changes the result of the adiabatic passage and, instead of the density-wave state, one obtains a disordered state. Intrinsic relation of the problem to Bloch oscillations of hard-core bosons in a 2D lattice is illuminated.
We investigate the resonantly enhanced tunneling dynamics of ultracold bosons loaded on a tilted 1-D optical lattice, which can be used to simulate a chain of Ising spins and associated quantum phase transitions. The center of mass motion after a sudden tilt both at commensurate and incommensurate fillings is obtained via analytic, time-dependent exact diagonalization and density matrix renormalization group methods (adaptive t-DMRG). We identify a maximum in the amplitude of the center of mass oscillations at the quantum critical point of the effective spin system. For the dynamics of incomme…
Bloch oscillations appear for a particle in a weakly tilted periodic potential. The intrinsic spin Hall effect is an outcome of a spin-orbit coupling. We demonstrate that both these phenomena can be realized simultaneously in a gas of weakly interacting ultracold atoms exposed to a tilted optical lattice and to a set of spatially dependent light fields inducing an effective spin-orbit coupling. It is found that both the spin Hall as well as the Bloch oscillation effects may coexist, showing, however, a strong correlation between the two. These correlations are manifested as a transverse spin c…
We show that and how ultra-cold atoms in an accelerated two-band lattice are a controlled realization of Landau--Zener--Stückelberg interferometry.
Dipole condensates in tilted Bose-Hubbard chains
本文通过场论分析与DMRG模拟研究了强倾斜光晶格中同时守恒玻色子数和偶极矩的Bose-Hubbard链，发现其相图不含可压缩相，而是由多种奇异偶极子凝聚体主导，并提出了冷原子实验识别方案。
Integrable 3-Site, Tilted, Extended Bose-Hubbard Model with Nearest-Neighbour Interactions
针对具有偶极相互作用的冷原子系统，提出一种可积的3位点扩展Bose-Hubbard模型，并通过Bethe ansatz方法得到能谱和本征态。
We simulate ultra-cold interacting Bosons in quasi-one-dimensional, incommensurate optical lattices. In the tight-binding limit, these lattices have pseudo-random on-site energies and thus can potentially lead to Anderson localization. We explore the parameter regimes that lead to Anderson localization and investigate the role of repulsive interactions, harmonic confinement and finite temperature. We find that interactions can obscure the exponential localization characteristic of Anderson localization, thus impeding the direct observation of this phenomenon when interactions are present.
We propose a scheme for a single-atom quantum heat engine based on ultra-cold atom technologies. Building on the high degree of control typical of cold atom systems, we demonstrate that three paradigmatic heat engines -- Carnot, Otto and Diesel -- are within reach of state-of-the-art technology, and their performances can be benchmarked experimentally. We discuss the implementation of these engines using realistic parameters and considering the friction effects that limit the maximum obtainable performances in real-life experiments. We further consider the use of super-adiabatic transformation…
We propose a realistic experiment to demonstrate a dynamic Kosterlitz-Thouless transition in ultra-cold atomic gases in two dimensions. With a numerical implementation of the Truncated Wigner Approximation we simulate the time evolution of several correlation functions, which can be measured via matter wave interference. We demonstrate that the relaxational dynamics is well-described by a real-time renormalization group approach, and argue that these experiments can guide the development of a theoretical framework for the understanding of critical dynamics.
Stability of superfluids in tilted optical lattices with periodic driving
本研究实验探究周期驱动倾斜光晶格最低能带中超流体的声子模式与相互作用驱动不稳定性，确定稳定/不稳定参数区并给出通用共振条件，利用微运动分析声子增长，为强相互作用驱动系统的量子模拟提供稳定参数预测依据。
Non-Fermi liquids from kinetic constraints in tilted optical lattices
本文研究倾斜光晶格中同时守恒粒子数和质心的动力学约束费米-哈伯德模型，发现其可稳定由费米子与无能隙玻色场耦合描述的非费米液体相，为超冷原子平台研究非费米液体开辟新途径。
We demonstrate clear collective atomic recoil motion in a dilute, momentum-squeezed, ultra-cold degenerate fermion gas by circumventing the effects of Pauli blocking. Although gain from bosonic stimulation is necessarily absent because the quantum gas obeys Fermi-Dirac statistics, collective atomic recoil motion from the underlying wave-mixing process is clearly visible. With a single pump pulse of the proper polarization, we observe two mutually-perpendicular wave-mixing processes occurring simultaneously. Our experiments also indicate that the red-blue pump detuning asymmetry observed with B…
Ultra cold fermionic atoms placed in a synthetic magnetic field arrange themselves in Landau levels. We theoretically study the optomechanical interaction between the light field and collective excitations of such fermionic atoms in synthetic magnetic field by placing them in side a Fabry Perot cavity. We derive the effective hamiltonian for particle hole excitations from a filled Landau level using a bosonization technique and obtain an expression for the cavity transmission spectrum. Using this we show that the cavity transmission spectrum demonstrates cold atom analogue of Subnikov de Hass …
Density-wave-type supersolid of two-dimensional tilted dipolar bosons
本研究预测二维倾斜偶极玻色子气体在零温下存在稳定的密度波型超固态相，该相表现为条纹、弹性、玻色-爱因斯坦凝聚与超流性共存，并发现随倾角增大相变类型从一阶变为二阶。
We study cold atomic gases with a contact interaction and confined into one-dimension. Crossing the confinement induced resonance the correlation between the bosons increases, and introduces an effective range for the interaction potential. Using the mapping onto the sine-Gordon model and a Hubbard model in the strongly interacting regime allows us to derive the phase diagram in the presence of an optical lattice. We demonstrate the appearance of a phase transition from a Luttinger liquid with algebraic correlations into a crystalline phase with a particle on every second lattice site.
System of a two-flavor mixture of ultra-cold fermions confined in a one-dimensional harmonic trap is studied in the frame of the center of mass. We present a numerical method of obtaining energetic spectra in this frame for an arbitrary mass ratio of fermionic species. We identify a specific invariant encoded in many-body correlations which enable one to determine an eigenstate of the Hamiltonian and to label excitations of the center of mass. The tool presented may be particularly useful in experimental analysis of the interparticle interactions which do not affect the center of mass excitati…
Motivated by recent experiments with ultracold magnetic atoms trapped in optical lattices where the orientation of atomic dipoles can be fully controlled by external fields, we study the ground state properties of dipolar bosons trapped in a two-dimensional lattice with density-induced hopping and where the dipoles are tilted along the $xz$ plane. We present ground state phase diagrams of the above system at different tilt angles. We find that, as the dipolar interaction increases, the superfluid phase at half filling factor is destroyed in favor of either a checkerboard or stripe solid phase …
We show that the general quantum state of synthetically spin-orbit coupled ultra cold bosonic atom whose condensate was experimentally created recently ( Y. J. Lin {\it et al.}, Nature, {\bf 471}, 83, (2011)), shows entanglement between motional degrees of freedom ( momentum) and internal degrees of freedom (hyperfine spin). We demonstrate the violation of Bell-like inequality (CHSH) for such states that provides a unique opportunity to verify fundamental principle like quantum non-contextuality for commutating observables which are not spatially separated. We analyze in detail the Rabi oscill…
We study the behavior of excitations in the tilted one-dimensional Bose-Hubbard model. In the phase with broken symmetry, fundamental excitations are domain-walls which show fractional statistics. Using perturbation theory, we derive an analytic model for the time evolution of these fractional excitations, and demonstrate the existence of a repulsively bound state above a critical center of mass momentum. The validity of the perturbative analysis is confirmed by the use of t- DMRG simulations. These findings open the path for experimental detection of fractional particles in cold atomic gases.
We study the spectrum of density fluctuations in the ultra-cold gas of neutral atoms, confined in a magneto-optical trap. We determine the corresponding amplitude and spectra of laser light scattered by this medium. We derive an expression for the dynamical structure function, by using a test particle method. We propose to use the collective laser scattering as a diagnostic method for the microscopic properties of the ultra-cold matter. This will also allow us to check on the atomic correlations which are mediated by the collective mean field inside the gas.
We propose a scheme to implement an optical Kagome lattice for ultra-cold atoms with controllable s-wave interactions between nearest neighbor sites and a gauge potential. The atoms occupy three different internal atomic levels with electromagnetically-induced coupling between the levels. We show that by appropriately shifting the triangular lattice potentials, experienced by atoms in different levels, the Kagome lattice can be realized using only two standing waves, generating a highly frustrated quantum system for the atoms.
We present a general model allowing "quantum simulation" of one-dimensional Dirac models with 2- and 4-component spinors using ultracold atoms in driven 1D tilted optical latices. The resulting Dirac physics is illustrated by one of its well-known manifestations, Zitterbewegung. This general model can be extended and applied with great flexibility to more complex situations.
We present a method for amplifying a single or scattered impurities immersed in a background gas of ultra cold atoms so that they can be optically imaged and spatially resolved. Our approach relies on a Raman transfer between two stable atomic hyperfine states that is conditioned on the presence of an impurity atom. The amplification is based on the strong interaction among atoms excited to Rydberg states. We perform a detailed analytical study of the performance of the proposed scheme with particular emphasis on the influence of many-body effects.
Recent experiments with ultra-cold atoms have demonstrated the possibility of realizing experimentally fermionic superfluids with imbalanced spin populations. We discuss how these developments have shed a new light on a half- century old open problem in condensed matter physics, and raised new interrogations of their own.
The entropy of the coexisting gas of ultra cold fermionic atoms and Bosonic molecular condensate confined in a magnetic trap has been calculated from equation of motion approaches. We have found that the entropy production depends not only to the Feshbach resonance and but also under certain limits to Rabi type oscillation and Bose Josephson junction type oscillation.
This dissertation presents theoretical investigations of universality and finite-range corrections in few- and many-boson systems. The main focus is on ultra-cold atomic gases seen both from the three-body, many-body, and mean-field perspective. Topics include corrections to Efimov physics, the N-body Efimov effect, two-body correlations in condensates and higher-order interactions in mean-field BECs.
We start by reviewing the concept of gauge invariance in quantum mechanics, for Abelian and Non-Ableian cases. Then we idescribe how the various gauge potential and field can be associated with the geometrical phase acquired by a quantum mechanical wave function while adiabatically evolving in a parameter space. Subsequently we show how this concept is exploited to generate light induced gauge field for neutral ultra cold bosonic atoms. As an example of such light induced Abelian and Non Abelian gauge field for ultra cold atoms we disucss ultra cold atoms in a rotating trap and creation of syn…
We investigate an ultra-cold mixture of Bose gases interacting via spin-changing collisions by studying the dynamics of spin fluctuations. The experimental implementation employs $^{23}$Na and $^{7}$Li atoms, which are prepared out of equilibrium across a wide range of initial conditions. We identify three regimes in the dynamics of the system for different initial states: a long-lived metastable regime, an instability range with strong growth of fluctuations, and a fast relaxing regime approaching thermal equilibrium. Theoretical modelling of the data allows us to reconstruct effective potent…
We discuss the possibility of realizing a non-Hermitian, i.e. an open two-well system of ultra-cold atoms by enclosing it with additional time-dependent wells that serve as particle reservoirs. With the appropriate design of the additional wells PT-symmetric currents can be induced to and from the inner wells, which support stable solutions. We show that interaction in the mean-field limit does not destroy this property. As a first method we use a simplified variational ansatz leading to a discrete nonlinear Schrödinger equation. A more accurate and more general variational ansatz is then used…
Solutions for the Nonlinear Schrodinger equation for collapsing Bose gas with attraction. This is a copy of the paper published in 1992 in Proceedings of NATO Advanced Research workshop on Singularities in Fluids, Plasmas and Optics (Heraklion, Greece) edited by R.E. Caflisch and G.C. Papanicolaou (Kluwer Academic).
We predict the effect of the roton instability for a two-dimensional weakly interacting gas of tilted dipoles in a single homogeneous quantum layer. Being typical for strongly correlated systems, the roton phenomena appear to occur in a weakly interacting gas. It is important that in contrast to a system of normal to wide layer dipoles, breaking of the rotational symmetry for a system of tilted dipoles leads to the convergence of the condensate depletion even up to the threshold of the roton instability, with mean-field approach being valid. Predicted effects can be observed in a wide class of…
We use optical tweezers based on time-multiplexed acousto-optic deflectors to trap ultra-cold cesium atoms in one-dimensional arrays of atomic ensembles. For temperatures between 2.5 $μ$K and 50 nK we study the maximal time between optical tweezer pulses that retains the number of atoms in a single trap. This time provides an estimate on the maximal number of sites in an array of time-multiplexed optical tweezers. We demonstrate evaporative cooling of atoms in arrays of up to 25 optical tweezer traps and the preparation of atoms in a box potential. Additionally, we demonstrate three different …
For ultra-cold atomic gases close to the critical temperature, there is a divergent correlation behavior within the critical regime. This divergent correlation behavior is the cornerstone of the universal behavior within the critical regime, e.g. the universal critical exponent for the same class with very different physical systems. It is still quite challenging to observe this divergent correlation behavior in experiments with ultra-cold atomic gases. Here we consider theoretically the interference effect of the critical atomic Bose gas by a Kapitza-Dirac scattering. We find that the Kapitza…
We study the equilibrium current density profiles of harmonically trapped ultra-cold Fermi gases in quantum Hall-like states that appear when the quasi-two-dimensional trap is set in fast rotation. The density profile of the gas (in the rotating reference frame) consists of incompressible strips of constant quantized density separated by compressible regions in which the density varies. Remarkably, we find that the atomic currents flow in opposite directions in the compressible and incompressible regions -- a prediction that should be amenable to experimental verification.
We study spinless bosons in a decorated square lattice with a near-diagonal tilt. The resonant subspace of the tilted Mott insulator is described by an effective Hamiltonian of frustrated quantum Ising spins on a non-bipartite lattice. This generalizes an earlier proposal for the unfrustrated quantum Ising model in one dimension which was realized in a recent experiment on ultracold $^{87}$Rb atoms in an optical lattice. Very close to diagonal tilt, we find a quantum liquid state which is continuously connected to the paramagnet. Frustration can be reduced by increasing the tilt angle away fro…
We show the possible excitation of a phonon laser instability in an ultra-cold atomic gas confined in a magneto-optical trap. Such an effect results from a negative Landau damping of the collective density perturbations in the gas, leading to the coherent emission of phonons. This laser instability can be driven by a blue-detuned laser superimposed to the usual red-detuning laser beams which usually provide the cooling mechanism. Threshold conditions, instability growth rates and saturation levels are derived. This work generalizes, on theoretical grounds, the recent results obtained with sing…
We discuss the evolution from BCS to Bose superfluidity for ultracold fermions in two-dimensions and in the presence of simultaneous spin-orbit and Zeeman fields. We analyze several thermodynamic properties to characterize different superfluid phases including pressure, compressibility, induced polarization, and spin susceptibility. Furthermore, we compute the momentum distribution and construct topological invariants for each of the superfluid phases.
Ultra-cold bosons in zig-zag optical lattices present a rich physics due to the interplay between frustration, induced by lattice geometry, two-body interaction and three-body constraint. Unconstrained bosons may develop chiral superfluidity and a Mott-insulator even at vanishingly small interactions. Bosons with a three-body constraint allow for a Haldane-insulator phase in non-polar gases, as well as pair-superfluidity and density wave phases for attractive interactions. These phases may be created and detected within the current state of the art techniques.
We study dynamics of an ultra-cold atomic BCS superfluid driven towards the BCS superfluid-Fermi liquid quantum critical point by a gradual decrease of the pairing interaction. We analyze how the BCS superfluid falls out of equilibrium and show that the non-equilibrium gap and Cooper pair size reflect critical properties of the transition. We observe three stages of evolution: adiabatic where the Cooper pair size is inversely proportional to the equilibrium gap, weakly non-equilibrium where it is inversely proportional to the non-equilibrium gap, and strongly non-equilibrium where it decouples…
We investigate the Zel'dovich effect in the context of ultra-cold, harmonically trapped quantum gases. We suggest that currently available experimental techniques in cold-atoms research offer an exciting opportunity for a direct observation of the Zel'dovich effect without the difficulties imposed by conventional condensed matter and nuclear physics studies. We also demonstrate an interesting scaling symmetry in the level rearragements which has heretofore gone unnoticed.
Cold atom systems offer a great potential for the future design of new mesoscopic quantum systems with properties that are fundamentally different from semiconductor nanostructures, such as quantum dots and quantum wires with electrons. Here, we investigate the analog of a quantum wire using ultra-cold particles, and find a new scenario for the quantum transport: Attractive interactions may lead to a complete suppression of current in the low-bias range, a total current blockade. We demonstrate this effect for the example of ultra-cold quantum gases with dipolar interactions.
We present a construction of an improved two-mode model for modeling the dynamics of interacting ultra-cold bosons confined in a one-dimensional double well trap. Unlike in the typically used two-mode model based on the lowest single-particle eigenstates of the external potential, the improved model uses a basis of properly chosen effective wave functions originating in the many-body model. Accuracy of the improved model is examined and it is shown that within a certain limit of inter-particle interaction strength, the model recovers an exact evolution of the wells' populations much more close…
In this paper, we consider the transmission characteristics of an optical cavity loaded with ultra cold atoms in a one dimensional optical lattice at absolute zero temperature. In particular, we consider the situation when the many body quantum state of the ultra cold atoms is an insulating state with fixed number of atoms at each site, which can be either density wave (DW) or Mott insulator (MI) phase, each showing different type of discrete lattice translational symmetry. We provide a general framework of understanding the transmission spectrum from a single and two cavities/modes loaded wit…
An ultra-cold molecular plasma with extraordinarily long lifetime (~0.5 ms) is generated under strong collisional cooling conditions in the expansion region of a seeded supersonic jet expansion close to the nozzle. A resonant two-photon one-color laser process excites para-difluorobenzene molecules into the high-n Rydberg threshold region. Disorder heating during plasma formation is quenched by the high collision rate in the expansion, which keeps the ions at the translational jet temperature of 0.2K-0.7K. The Coulomb coupling parameter \Gammai for the ions is expected to be ca. 230-820.
Exact dynamics of two ultra-cold bosons confined in a one-dimensional double-well potential
该研究比较了一维双阱中两个超冷玻色子的精确动力学与双模模型近似，发现强相互作用下双模模型失效，需考虑更高单粒子态。
Quantum decay of a relativistic scalar field from a false vacuum is a fundamental idea in quantum field theory. It is relevant to models of the early Universe, where the nucleation of bubbles gives rise to an inflationary universe and the creation of matter. Here we propose a laboratory test using an experimental model of an ultra-cold spinor Bose gas. A false vacuum for the relative phase of two spin components, serving as the unstable scalar field, is generated by means of a modulated radio-frequency coupling of the spin components. Numerical simulations demonstrate the spontaneous formation…
Anomalous localization in spin chains with tilted interactions
该研究针对格点规范理论量子模拟器衍生的长程相互作用问题，分析了相互作用强度线性增长的自旋链的局域化性质，通过精确对角化发现局域化体积随相互作用程非单调变化，为相关量子模拟和长程系统的禁闭现象研究提供参考。
There has been a recent surge of interest and progress in creating subwavelength free-space optical potentials for ultra-cold atoms. A key open question is whether geometric potentials, which are repulsive and ubiquitous in the creation of subwavelength free-space potentials, forbid the creation of narrow traps with long lifetimes. Here, we show that it is possible to create such traps. We propose two schemes for realizing subwavelength traps and demonstrate their superiority over existing proposals. We analyze the lifetime of atoms in such traps and show that long-lived bound states are possi…
In this paper, by treating the hopping parameter in Bose-Hubbard model as a perturbation, with the help of the re-summed Green's function method and cumulants expansion, the momentum distribution function of the ultra-cold Bose system in triangular optical lattice is calculated analytically. By utilizing it, the time-of-flight absorption picture is plotted and the corresponding visibility is determined. The comparison between our analytical results and the experimental data from Ref.[4] exhibits a qualitative agreement.
We show that static and oscillating photon bubbles can be excited by diffused light in the laser cooled matter confined in a magneto-optical trap (MOT). The bubble instability is due to the coupling between the radiation field and the mean field oscillations of the ultra-cold gas, and it can provide a source for low frequency turbulence. We consider a diffusion dominated regime, which can be described by a radiation transport equation, coupled with the mean field equations for the cold atom gas. A perturbative analysis shows the occurrence of two different regimes with either oscillating or pu…
Advances in controlling and measuring systems of ultra-cold atoms provided strong motivation to theoretical investigations of quantum dynamics in closed many-body systems. Fundamental questions on quantum dynamics and statistical mechanics are now within experimental reach: How is thermalization achieved in a closed quantum system? How does quantum dynamics cross over to effective classical physics? Can such a thermal or classical fate be evaded? In these lectures, given at the Les Houches Summer School of Physics "Strongly Interacting Quantum Systems Out of Equilibrium", I introduce the stude…
This paper is a short introduction to cold atom physics and Bose-Einstein condensation. Light forces on atoms are presented, together with laser cooling, and a few atom traps: the magneto-optical trap, dipole traps and magnetic traps. A brief description of Bose-Einstein condensation is given together with some important links with condensed matter physics. The reader is referred to comprehensive reviews and to other lecture notes for further details on atom cooling, trapping and Bose-Einstein condensation.
The multiple scattering of light in a gas of ultra cold atoms is responsible for many exciting features observed in magneto-optical traps including the collective behavior forced by a Coulomb like potential. This field also induces plasma like phenomena in the cloud which allows the treatment of the system as a one component trapped plasma. With a fluid description and casting the thermodynamical behavior in the form of a polytropic equation of state we investigate the equilibrium profiles of rotating clouds and its dependence on the experiment characteristics. Numerical solutions predict the …
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