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A fundamental challenge in modern physics is bridging the gap between an exact description of few-body microscopic systems and the emergent description of many-body macroscopic systems. This gap may be bridged by tracing the properties of well-controlled intermediate-size mesoscopic systems. Here, we study mesoscopic signatures of a spatial self-organization phase transition in deterministically prepared arrays of between 10 and 22 atoms inside an optical cavity. Through precise engineering of the atom-cavity interactions, we reveal how critical behavior depends on atom number. We identify characteristic dynamical behaviors related to symmetry breaking and system size in the self-organized regime, and observe a finite optomechanical susceptibility at the critical point. This work opens the door to probing particle-number- and time-resolved properties of phase transitions in mesoscopic systems., Comment: 21 pages, 9 figures

We produce beams of neutral titanium (Ti) atoms in their metastable $3d^{3}(^4F){4}s$ $a^5F_5$ state by laser ablation into He, N$_2$, and Ar buffer gases. The high temperatures realized in the ablation process populate the $a^5F_5$ level without the need for optical pumping. Remarkably, we observe that Ti atoms in the $a^5F_5$ state survive $1000$'s of collisions with He and Ar buffer gas atoms without being quenched to lower-energy states. We study the yield of Ti atoms when ablated into buffer gases of varying species and pressure, quantify quenching rates and diffusion cross sections based on simple models, and provide insight into optimal design parameters for an ablation cell. Using a $3.3\,$cm ablation cell with interchangeable exit apertures, we produce metastable atom beams and quantify their brilliance and velocity distributions as functions of buffer gas pressure.

We present a theoretical framework for 'collective transition quenching', a quantum many-body dissipative phenomenon that occurs in systems with multiple collective decay channels. Despite the competition, interactions suppress all but the dominant decay transition, leading to a 'winner takes all' dynamic where the system primarily settles into the dominant ground state. We prove that, in the presence of permutation symmetry, this problem is exactly solvable for any number of competing channels. Additionally, we develop an approximate model for the dynamics by mapping the evolution into a continuity equation for a fluid, and show analytically that the dominant transition ratio converges to unity with increasing system size as a power-law, for any branching ratio. This near-deterministic preparation of the dominant ground state has broad applicability. As an example we discuss a protocol for molecular photoassociation where collective dynamics effectively acts as a catalyst, amplifying the yield in a particular final state. Our results open new avenues for many-body strategies in the preparation and control of quantum systems., Comment: 5 pages, 3 figures

We generate an atomic beam of titanium (Ti) using a "Ti-ball" Ti-sublimation pump, which is a common getter pump used in ultrahigh vacuum (UHV) systems. We show that the sublimated atomic beam can be optically pumped into the metastable $3d^{3}(^4F){4}s$ $a^5F_5$ state, which is the lower energy level in a nearly cycling optical transition that can be used for laser cooling. We measure the atomic density and transverse and longitudinal velocity distributions of the beam through laser fluorescence spectroscopy. We find a metastable atomic flux density of $4.3(2)\times10^9\,$s$^{-1}$cm$^{-2}$ with mean forward velocity $773(8)\,$m/s at $2.55\,$cm directly downstream of the center of the Ti-ball. Owing to the details of optical pumping, the beam is highly collimated along the transverse axis parallel to the optical pumping beam and the flux density falls off as $1/r$. We discuss how this source can be used to load atoms into a magneto-optical trap.

Atoms are not two-level systems, and their rich internal structure often leads to complex phenomena in the presence of light. Here, we analyze off-resonant light scattering including the full hyperfine and magnetic structure. We find a set of frequency detunings where the induced atomic dipole is the same irrespective of the Zeeman state, and where two-photon transitions that alter the atomic state turn off. For alkali atoms and alkaline-earth ions, if the hyperfine splitting is dominated by the magnetic dipole moment contribution, these detunings approximately coincide. Therefore, at a given ``magical'' detuning, all Zeeman states in a hyperfine manifold behave almost identically, and can be traced out to good approximation. This feature prevents state decoherence due to light scattering, which impacts quantum optics experiments and quantum information applications., Comment: 12+9 pages, 4+1 figures, 2 tables. This version is in line with the published version

We realize collective enhancement and suppression of light scattered by an array of tweezer-trapped $^{87}$Rb atoms positioned within a strongly coupled Fabry-P\'{e}rot optical cavity. We illuminate the array with light directed transverse to the cavity axis, in the low saturation regime, and detect photons scattered into the cavity. For an array with integer-optical-wavelength spacing each atom scatters light into the cavity with nearly identical scattering amplitude, leading to an observed $N^2$ scaling of cavity photon number as the atom number increases stepwise from $N=1$ to $N=8$. By contrast, for an array with half-integer-wavelength spacing, destructive interference of scattering amplitudes yields a non-monotonic, sub-radiant cavity intensity versus $N$. By analyzing the polarization of light emitted from the cavity, we find that Rayleigh scattering can be collectively enhanced or suppressed with respect to Raman scattering. We observe also that atom-induced shifts and broadenings of the cavity resonance are precisely tuned by varying the atom number and positions. Altogether, tweezer arrays provide exquisite control of atomic cavity QED spanning from the single- to the many-body regime.

We report out-of-equilibrium stabilization of the collective spin of an atomic ensemble through autonomous feedback by a driven optical cavity. For a magnetic field applied at an angle to the cavity axis, dispersive coupling to the cavity provides sensitivity to a combination of the longitudinal and transverse spin. Coherent backaction by cavity light onto the atoms, conditioned by the optical cavity susceptibility, stabilizes the collective spin state at an arbitrary energy. The set point tracking and closed-loop gain spectrum of the feedback system are characterized and found to agree closely with analytic predictions.

We propose an optical clock based on narrow, spin-forbidden M1 and E2 transitions in laser-cooled neutral titanium. These transitions exhibit much smaller black body radiation shifts than those in alkaline earth atoms, small quadratic Zeeman shifts, and have wavelengths in the S, C, and L-bands of fiber-optic telecommunication standards, allowing for integration with robust laser technology. We calculate lifetimes; transition matrix elements; dynamic scalar, vector, and tensor polarizabilities; and black body radiation shifts of the clock transitions using a high-precision relativistic hybrid method that combines a configuration interaction and coupled cluster approaches. We also calculate the line strengths and branching ratios of the transitions used for laser cooling. To identify magic trapping wavelengths, we have completed the largest-to-date direct dynamical polarizability calculations. Finally, we identify new challenges that arise in precision measurements due to magnetic dipole-dipole interactions and describe an approach to overcome them. Direct access to a telecommunications-band atomic frequency standard will aid the deployment of optical clock networks and clock comparisons over long distances., Comment: 5 pages, 2 figures main text; 8 pages, 3 figures supplementary text

Subsystem readout during a quantum process, or mid-circuit measurement, is crucial for error correction in quantum computation, simulation, and metrology. Ideal mid-circuit measurement should be faster than the decoherence of the system, high-fidelity, and nondestructive to the unmeasured qubits. Here, we use a strongly coupled optical cavity to read out the state of a single tweezer-trapped 87Rb atom within a small tweezer array. Measuring either atomic fluorescence or the transmission of light through the cavity, we detect both the presence and the state of an atom in the tweezer, within only tens of microseconds, with state preparation and measurement infidelities of roughly 0.5% and atom loss probabilities of around 1%. Using a two-tweezer system, we find measurement on one atom within the cavity causes no observable hyperfine-state decoherence on a second atom located tens of microns from the cavity volume. This high-fidelity mid-circuit readout method is a substantial step towards quantum error correction in neutral atom arrays., Comment: E.D. and Y.-H.L. contributed equally to this manuscript. Accepted for publication in Physical Review Letters

We analytically identify a new class of quantum scars protected by spatiotemporal translation symmetries, dubbed Floquet-Bloch scars. They distinguish from previous (quasi-)static scars by a rigid spectral pairing only possible in Floquet systems, where strong interaction and drivings equalize the quasienergy corrections to all scars and maintain their spectral spacings against generic bilinear perturbations. Scars then enforce the spatial localization and rigid discrete time crystal (DTC) oscillations as verified numerically in a trimerized kagome lattice model relevant to recent cold atom experiments. Our analytical solutions offer a potential scheme to understand the mechanisms for more generic translation-invariant DTCs., Comment: 6+11 pages

We realize a scanning probe microscope using single trapped $^{87}$Rb atoms to measure optical fields with subwavelength spatial resolution. Our microscope operates by detecting fluorescence from a single atom driven by near-resonant light and determining the ac Stark shift of an atomic transition from other local optical fields via the change in the fluorescence rate. We benchmark the microscope by measuring two standing-wave Gaussian modes of a Fabry-P\'{e}rot resonator with optical wavelengths of 1560 nm and 781 nm. We attain a spatial resolution of 300 nm, which is superresolving compared to the limit set by the 780 nm wavelength of the detected light. Sensitivity to short length scale features is enhanced by adapting the sensor to characterize an optical field via the force it exerts on the atom., Comment: 5 pages, 4 figures; Accepted for publication in Phys. Rev. Lett

The band structure of a crystal may have points where two or more bands are degenerate in energy and where the geometry of the Bloch state manifold is singular, with consequences for material and transport properties. Ultracold atoms in optical lattices have been used to characterize such points only indirectly, e.g., by detection of an Abelian Berry phase, and only at singularities with linear dispersion (Dirac points). Here, we probe band-structure singularities through the non-Abelian transformation produced by transport directly through the singular points. We prepare atoms in one Bloch band, accelerate them along a quasi-momentum trajectory that enters, turns, and then exits the singularities at linear and quadratic touching points of a honeycomb lattice. Measurements of the band populations after transport identify the winding numbers of these singularities to be 1 and 2, respectively. Our work opens the study of quadratic band touching points in ultracold-atom quantum simulators, and also provides a novel method for probing other band geometry singularities.

The fluctuations in thermodynamic and transport properties in many-body systems gain importance as the number of constituent particles is reduced. Ultracold atomic gases provide a clean setting for the study of mesoscopic systems; however, the detection of temporal fluctuations is hindered by the typically destructive detection, precluding repeated precise measurements on the same sample. Here, we overcome this hindrance by utilizing the enhanced light--matter coupling in an optical cavity to perform a minimally invasive continuous measurement and track the time evolution of the atom number in a quasi two-dimensional atomic gas during evaporation from a tilted trapping potential. We demonstrate sufficient measurement precision to detect atom number fluctuations well below the level set by Poissonian statistics. Furthermore, we characterize the non-linearity of the evaporation process and the inherent fluctuations of the transport of atoms out of the trapping volume through two-time correlations of the atom number. Our results establish coupled atom--cavity systems as a novel testbed for observing thermodynamics and transport phenomena in mesosopic cold atomic gases and, generally, pave the way for measuring multi-time correlation functions of ultracold quantum gases., Comment: Significantly extended discussion of Fig. 4. Accepted for publication in Phys. Rev. X

We measure and analyze the isotope shifts the multiplet of transitions between the metastable a$^5$F and excited y$^5$G$^\circ$ terms of neutral titanium by probing a titanium vapor in a hollow cathode lamp using saturated absorption spectroscopy. We resolve the five $J\to J+1$ and the four $J\to J$ transitions within the multiplet for each of the the three $I=0$ stable isotopes ($^{46}$Ti, $^{48}$Ti, and $^{50}$Ti). The isotope shifts on these transitions allow us to determine the isotope-dependent variation in the fine-structure splitting of the a$^5$F and y$^5$G$^\circ$ levels themselves. Combined with existing knowledge of the nuclear charge radii of titanium nuclei, we derive the specific mass and field shifts, which arise from correlated electronic motion and electronic density at the nucleus respectively, and further observe a strong $J$-dependent variation in each. Our results yield insight into the electronic and nuclear structure of transition metal atoms like titanium, and also characterize optical transitions that may allow for optical manipulation of ultracold gases of transition metal species., Comment: 7 pages, 6 figures

We propose the application of laser cooling to a number of transition-metal atoms, allowing numerous bosonic and fermionic atomic gases to be cooled to ultra-low temperatures. The non-zero electron orbital angular momentum of these atoms implies that strongly atom-state-dependent light-atom interactions occur even for light that is far-detuned from atomic transitions. At the same time, many transition-metal atoms have small magnetic dipole moments in their low-energy states, reducing the rate of dipolar-relaxation collisions. Altogether, these features provide compelling opportunities for future ultracold-atom research. Focusing on the case of atomic titanium, we identify the metastable $a ^5F_5$ state as supporting a $J \rightarrow J+1$ optical transition with properties similar to the D2 transition of alkali atoms, and suited for laser cooling. The high total angular momentum and electron spin of this state suppresses leakage out of the the nearly closed optical transition to a branching ratio estimated below $\sim 10^{-5}$. Following the pattern exemplified by titanium, we identify optical transitions that are suited for laser cooling of elements in the scandium group (Sc, Y, La), the titanium group (Ti, Zr), the vanadium group (V, Nb), the manganese group (Mn, Tc), and the iron group (Fe, Ru)., Comment: 12 pages, 6 figures

Geometric frustration of particle motion in a kagome lattice causes the single-particle band structure to have a flat s-orbital band. We probe this band structure by exciting a Bose-Einstein condensate into excited Bloch states of an optical kagome lattice, and then measuring the group velocity through the atomic momentum distribution. We find that interactions renormalize the band structure of the kagome lattice, greatly increasing the dispersion of the third band that, according to non-interacting band theory, should be nearly non-dispersing. Measurements at various lattice depths and gas densities agree quantitatively with predictions of the lattice Gross-Pitaevskii equation, indicating that the observed distortion of band structure is caused by the disortion of the overall lattice potential away from the kagome geometry by interactions.

We observe spin transfer within a non-degenerate heteronuclear spinor atomic gas comprised of a small $^7$Li population admixed with a $^{87}$Rb bath, with both elements in their $F=1$ hyperfine spin manifolds and at temperatures of 10's of $\mu$K. Prepared in a non-equilibrium initial state, the $^7$Li spin distribution evolves through incoherent spin-changing collisions toward a steady-state distribution. We identify and measure the cross-sections of all three types of spin-dependent heteronuclear collisions, namely the spin-exchange, spin-mixing, and quadrupole-exchange interactions, and find agreement with predictions of heteronuclear $^7$Li-$^{87}$Rb interactions at low energy. Moreover, we observe that the steady state of the $^7$Li spinor gas can be controlled by varying the composition of the $^{87}$Rb spin bath with which it interacts.

Efficient and versatile interfaces for the interaction of light with matter are an essential cornerstone for quantum science. A fundamentally new avenue of controlling light-matter interactions has been recently proposed based on the rich interplay of photon-mediated dipole-dipole interactions in structured subwavelength arrays of quantum emitters. Here we report on the direct observation of the cooperative subradiant response of a two-dimensional (2d) square array of atoms in an optical lattice. We observe a spectral narrowing of the collective atomic response well below the quantum-limited decay of individual atoms into free space. Through spatially resolved spectroscopic measurements, we show that the array acts as an efficient mirror formed by only a single monolayer of a few hundred atoms. By tuning the atom density in the array and by changing the ordering of the particles, we are able to control the cooperative response of the array and elucidate the interplay of spatial order and dipolar interactions for the collective properties of the ensemble. Bloch oscillations of the atoms out of the array enable us to dynamically control the reflectivity of the atomic mirror. Our work demonstrates efficient optical metamaterial engineering based on structured ensembles of atoms and paves the way towards the controlled many-body physics with light and novel light-matter interfaces at the single quantum level., Comment: 8 pages, 5 figures + 12 pages Supplementary Infomation

Microgravity eases several constraints limiting experiments with ultracold and condensed atoms on ground. It enables extended times of flight without suspension and eliminates the gravitational sag for trapped atoms. These advantages motivated numerous initiatives to adapt and operate experimental setups on microgravity platforms. We describe the design of the payload, motivations for design choices, and capabilities of the Bose-Einstein Condensate and Cold Atom Laboratory (BECCAL), a NASA-DLR collaboration. BECCAL builds on the heritage of previous devices operated in microgravity, features rubidium and potassium, multiple options for magnetic and optical trapping, different methods for coherent manipulation, and will offer new perspectives for experiments on quantum optics, atom optics, and atom interferometry in the unique microgravity environment on board the International Space Station.

We measure the interspecies interaction strength between $^7$Li and $^{87}$Rb atoms through cross-dimensional relaxation of two-element gas mixtures trapped in a spherical quadrupole magnetic trap. We record the relaxation of an initial momentum-space anisotropy in a lithium gas when co-trapped with rubidium atoms, with both species in the $|F=1, m_F = -1\rangle$ hyperfine state. Our measurements are calibrated by observing cross-dimensional relaxation of a $^{87}$Rb-only trapped gas. Through Monte Carlo simulations, we compare the observed relaxation to that expected given the theoretically predicted energy-dependent differential cross section for $^7$Li-$^{87}$Rb collisions. The experimentally observed relaxation occurs significantly faster than predicted theoretically, a deviation that appears incompatible with other experimental data characterising the $^7$Li-$^{87}$Rb molecular potential.

We produce a trimerized kagome lattice for ultracold atoms using an optical superlattice formed by overlaying triangular lattices generated with two colors of light at a 2:1 wavelength ratio. Adjusting the depth of each lattice tunes the strong intra-trimer (J) and weak inter-trimer (J') tunneling energies, and also the on-site interaction energy U. Two different trimerization patterns are distinguished using matter-wave diffraction. We characterize the coherence of a strongly interacting Bose gas in this lattice, observing persistent nearest-neighbor spatial coherence in the large U/J' limit, and that such coherence displays asymmetry between the strongly and the weakly coupled bonds.

Generation and manipulation of many-body entangled states is of considerable interest, for applications in quantum simulation or sensing, for example. Measurement and verification of the resulting many-body state presents a formidable challenge, however, which can be simplified by multiplexed readout using shared measurement resources. In this work, we analyze and demonstrate state retrodiction for a system of optomechanical oscillators coupled to a single-mode optical cavity. Coupling to the shared cavity field facilitates simultaneous optical measurement of the oscillators' transient dynamics at distinct frequencies. Optimal estimators for the oscillators' initial state can be defined as a set of linear matched filters, derived from a detailed model for the detected homodyne signal. We find that the optimal state estimate for optomechanical retrodiction is obtained from high-cooperativity measurements, reaching estimate sensitivity at the Standard Quantum Limit (SQL). Simultaneous estimation of the state of multiple oscillators places additional limits on the estimate precision, due to the diffusive noise each oscillator adds to the optomechanical signal. However, we show that the sensitivity of simultaneous multi-mode state retrodiction reaches the SQL for sufficiently well-resolved oscillators. Finally, an experimental demonstration of two-mode retrodiction is presented, which requires further accounting for technical fluctuations of the oscillator frequency., Comment: 13 pages, 6 figures

A microscopic understanding of molecules is essential for many fields of natural sciences but their tiny size hinders direct optical access to their constituents. Rydberg macrodimers - bound states of two highly-excited Rydberg atoms - feature bond lengths easily exceeding optical wavelengths. Here we report on the direct microscopic observation and detailed characterization of such macrodimers in a gas of ultracold atoms in an optical lattice. The size of about 0.7 micrometers, comparable to the size of small bacteria, matches the diagonal distance of the lattice. By exciting pairs in the initial two-dimensional atom array, we resolve more than 50 vibrational resonances. Using our spatially resolved detection, we observe the macrodimers by correlated atom loss and demonstrate control of the molecular alignment by the choice of the vibrational state. Our results allow for precision testing of Rydberg interaction potentials and establish quantum gas microscopy as a powerful new tool for quantum chemistry., Comment: 13 pages, 9 figures

We realize a spin-orbit interaction between the collective spin precession and center-of-mass motion of a trapped ultracold atomic gas, mediated by spin- and position-dependent dispersive coupling to a driven optical cavity. The collective spin, precessing near its highest-energy state in an applied magnetic field, can be approximated as a negative-mass harmonic oscillator. When the Larmor precession and mechanical motion are nearly resonant, cavity mediated coupling leads to a negative-mass instability, driving exponential growth of a correlated mode of the hybrid system. We observe this growth imprinted on modulations of the cavity field and estimate the full covariance of the resulting two-mode state by observing its transient decay during subsequent free evolution., Comment: 5 pages, 4 figures, and supplementary material

The mean-field treatment of the Bose-Hubbard model predicts properties of lattice-trapped gases to be insensitive to the specific lattice geometry once system energies are scaled by the lattice coordination number $z$. We test this scaling directly by comparing coherence properties of $^{87}$Rb gases that are driven across the superfluid to Mott insulator transition within optical lattices of either the kagome ($z=4$) or the triangular ($z=6$) geometries. The coherent fraction measured for atoms in the kagome lattice is lower than for those in a triangular lattice with the same interaction and tunneling energies. A comparison of measurements from both lattices agrees quantitatively with the scaling prediction. We also study the response of the gas to a change in lattice geometry, and observe the dynamics as a strongly interacting kagome-lattice gas is suddenly "hole-doped" by introducing the additional sites of the triangular lattice.

We analyze a method for preparing low-entropy many-body states in isolated quantum optical systems of atoms, ions and molecules. Our approach is based upon shifting entropy between different regions of a system by spatially modulating the magnitude of the effective Hamiltonian. We conduct two case studies, on a topological spin chain and the spinful fermionic Hubbard model, focusing on the key question: can a "conformal cooling quench" remove sufficient entropy within experimentally accessible timescales? Finite temperature, time-dependent matrix product state calculations reveal that even moderately sized "bath" regions can remove enough energy and entropy density to expose coherent low temperature physics. The protocol is particularly natural in systems with long-range interactions such lattice-trapped polar molecules and Rydberg dressed atoms where the magnitude of the Hamiltonian scales directly with the density. To this end, we propose a simple implementation of conformal cooling quenches in a dilutely-filled optical lattice, where signatures of quantum magnetism can be observed., Comment: 10 pages, 9 figures. V2: Modified discussion of experimental implementation and added references

We demonstrate continuous measurement and coherent control of the collective spin of an atomic ensemble undergoing Larmor precession in a high-finesse optical cavity. The coupling of the precessing spin to the cavity field yields phenomena similar to those observed in cavity optomechanics, including cavity amplification, damping, and optical spring shifts. These effects arise from autonomous optical feedback onto the atomic spin dynamics, conditioned by the cavity spectrum. We use this feedback to stabilize the spin in either its high- or low-energy state, where, in equilibrium with measurement back-action heating, it achieves a steady-state temperature, indicated by an asymmetry between the Stokes and anti-Stokes scattering rates. For sufficiently large Larmor frequency, such feedback stabilizes the spin ensemble in a nearly pure quantum state, in spite of continuous measurement by the cavity field., Comment: 5 pages, 4 figures, and supplemental material

Out-of-time-order correlation functions provide a proxy for diagnosing chaos in quantum systems. We propose and analyze an interferometric scheme for their measurement, using only local quantum control and no reverse time evolution. Our approach utilizes a combination of Ramsey interferometry and the recently demonstrated ability to directly measure Renyi entropies. To implement our scheme, we present a pair of cold-atom-based experimental blueprints; moreover, we demonstrate that within these systems, one can naturally realize the transverse-field Sherrington-Kirkpatrick (TFSK) model, which exhibits certain similarities with fast scrambling black holes. We perform a detailed numerical study of scrambling in the TFSK model, observing an interesting interplay between the fast scrambling bound and the onset of spin-glass order., Comment: 6 pages, 3 figures

While the existence of quantum superpositions of massive particles over microscopic separations has been established since the birth of quantum mechanics, the maintenance of superposition states over macroscopic separations is a subject of modern experimental tests. In Ref. [1], T. Kovachy et al. report on applying optical pulses to place a freely falling Bose-Einstein condensate into a superposition of two trajectories that separate by an impressive distance of 54 cm before being redirected toward one another. When the trajectories overlap, a final optical pulse produces interference with high contrast, but with random phase, between the two wave packets. Contrary to claims made in Ref. [1], we argue that the observed interference is consistent with, but does not prove, that the spatially separated atomic ensembles were in a quantum superposition state. Therefore, the persistence of such superposition states remains experimentally unestablished., Comment: Comment on Kovachy et al., "Quantum superposition at the half-metre scale," Nature 528, 530 (2015); see also reply by Kovachy et al., arXiv:1607.03485

We observe the condensation of magnon excitations within an $F=1$ $^{87}$Rb spinor Bose-Einstein condensed gas. Magnons are pumped into a longitudinally spin-polarized gas, allowed to equilibrate to a non-degenerate distribution, and then cooled evaporatively at near-constant net longitudinal magnetization whereupon they condense. We find magnon condensation to be described quantitatively as the condensation of free particles in an effective potential that is uniform within the ferromagnetic condensate volume, evidenced by the number and distribution of magnons at the condensation transition. Transverse magnetization images reveal directly the spontaneous, inhomogeneous symmetry breaking by the magnon quasi-condensate, including signatures of Mermin-Ho spin textures that appear as phase singularities in the magnon condensate wavefunction., Comment: 5 pages, 3 figures

In a spinor Bose-Einstein gas, the non-zero hyperfine spin of the gas becomes an accessible degree of freedom. At low temperature, such a gas shows both magnetic and superfluid order, and undergoes both density and spin dynamics. These lecture notes present a general overview of the properties of spinor Bose-Einstein gases. The notes are divided in five sections. In the first, we summarize basic properties of multi-component quantum fluids, focusing on the specific case of spinor Bose-Einstein gases and the role of rotational symmetry in defining their properties. Second, we consider the magnetic state of a spinor Bose-Einstein gas, highlighting effects of thermodynamics and Bose-Einstein statistics and also of spin-dependent interactions between atoms. In the third section, we discuss methods for measuring the properties of magnetically ordered quantum gases and present newly developed schemes for spin-dependent imaging. We then discuss the dynamics of spin mixing in which the spin composition of the gas evolves through the spin-dependent interactions within the gas. We discuss spin mixing first from a microscopic perspective, and then advance to discussing collective and beyond-mean-field dynamics. The fifth section reviews recent studies of the magnetic excitations of quantum-degenerate spinor Bose gases. We conclude with some perspectives on future directions for research., Comment: 70 pages, 22 figures. Lecture notes for the 2014 Enrico Fermi Summer School on Quantum Matter at Ultralow Temperatures

Quantum spin liquids are a new class of magnetic ground state in which spins are quantum mechanically entangled over macroscopic scales. Motivated by recent advances in the control of polar molecules, we show that dipolar interactions between S=1/2 moments stabilize spin liquids on the triangular and kagome lattices. In the latter case, the moments spontaneously break time-reversal, forming a chiral spin liquid with robust edge modes and emergent semions. We propose a simple route toward synthesizing a dipolar Heisenberg antiferromagnet from lattice-trapped polar molecules using only a single pair of rotational states and a constant electric field., Comment: 11 pages, 8 figures

Ultracold gases promise access to many-body quantum phenomena at convenient length and time scales. However, it is unclear whether the entropy of these gases is low enough to realize many phenomena relevant to condensed matter physics, such as quantum magnetism. Here we report reliable single-shot temperature measurements of a degenerate $^{87}$Rb gas by imaging the momentum distribution of thermalized magnons, which are spin excitations of the atomic gas. We record average temperatures as low as $0.022(1)_\text{stat}(2)_\text{sys}$ times the Bose-Einstein condensation temperature, indicating an entropy per particle, $S/N\approx0.001\, k_B$ at equilibrium, that is well below the critical entropy for antiferromagnetic ordering of a Bose-Hubbard system. The magnons themselves can reduce the temperature of the system by absorbing energy during thermalization and by enhancing evaporative cooling, allowing low-entropy gases to be produced within deep traps., Comment: 8 pages, 3 figures, including methods; Supplemental material: 2 pages, 1 figure

A complex quantum system can be constructed by coupling simple quantum elements to one another. For example, trapped-ion or superconducting quantum bits may be coupled by Coulomb interactions, mediated by the exchange of virtual photons. Alternatively quantum objects can be coupled by the exchange of real photons, particularly when driven within resonators that amplify interactions with a single electro-magnetic mode. However, in such an open system, the capacity of a coupling channel to convey quantum information or generate entanglement may be compromised. Here, we realize phase-coherent interactions between two spatially separated, near-ground-state mechanical oscillators within a driven optical cavity. We observe also the noise imparted by the optical coupling, which results in correlated mechanical fluctuations of the two oscillators. Achieving the quantum backaction dominated regime opens the door to numerous applications of cavity optomechanics with a complex mechanical system. Our results thereby illustrate the potential, and also the challenge, of coupling quantum objects with light., Comment: 25 pages, 6 figures

The dynamics of internal spin, electronic orbital, and nuclear motion states of atoms and molecules have preoccupied the atomic and molecular physics community for decades. Increasingly, such dynamics are being examined within many-body systems composed of atomic and molecular gases. Our findings sometimes bear close relation to phenomena observed in condensed-matter systems, while on other occasions they represent truly new areas of investigation. I discuss several examples of spin dynamics that occur within spinor Bose-Einstein gases, highlighting the advantages of spin-sensitive imaging for understanding and utilizing such dynamics., Comment: Chapter in upcoming Review Volume entitled "From Atomic to Mesoscale: The Role of Quantum Coherence in Systems of Various Complexities" from World Scientific

We measure the mass, gap, and magnetic moment of a magnon in the ferromagnetic $F=1$ spinor Bose-Einstein condensate of $^{87}$Rb. We find an unusually heavy magnon mass of $1.038(2)_\mathrm{stat}(8)_\mathrm{sys}$ times the atomic mass, as determined by interfering standing and running coherent magnon waves within the dense and trapped condensed gas. This measurement is shifted significantly from theoretical estimates. The magnon energy gap of $h\times 2.5(1)_\mathrm{stat}(2)_\mathrm{sys}\;\mathrm{Hz}$ and the effective magnetic moment of $-1.04(2)_\mathrm{stat}(8)\,\mu_\textrm{bare}$ times the atomic magnetic moment are consistent with mean-field predictions. The nonzero energy gap arises from magnetic dipole-dipole interactions., Comment: 5 pages, 3 figures

The Heisenberg uncertainty principle sets a lower bound on the sensitivity of continuous optical measurements of force. This bound, the standard quantum limit, can only be reached when a mechanical oscillator subjected to the force is unperturbed by its environment, and when measurement imprecision from photon shot-noise is balanced against disturbance from measurement backaction. We apply an external force to the center-of-mass motion of an ultracold atom cloud in a high-finesse optical cavity. The optomechanically transduced response clearly demonstrates the trade-off between measurement imprecision and back-action noise. We achieve a sensitivity that is consistent with theoretical predictions for the quantum limit given the atoms' slight residual thermal disturbance and the photodetection quantum efficiency, and is a factor of 4 above the absolute standard quantum limit., Comment: 17 pages, 4 figures

We create an ultracold-atoms-based cavity optomechanical system in which as many as six distinguishable mechanical oscillators are prepared, and optically detected, near their ground states of motion. We demonstrate that the motional state of one oscillator can be selectively addressed while preserving neighboring oscillators near their ground states to better than 95% per excitation quantum. We also show that our system offers nanometer-scale spatial resolution of each mechanical element via optomechanical imaging. This technique enables in-situ, parallel sensing of potential landscapes, a capability relevant to active research areas of atomic physics and force-field detection in optomechanics., Comment: 4 pages, 4 figures, and supplemental information

The precision of compact inertial sensing schemes using trapped- and guided-atom interferometers has been limited by uncontrolled phase errors caused by trapping potentials and interactions. Here, we propose an acoustic interferometer that uses sound waves in a toroidal Bose-Einstein condensate to measure rotation, and we demonstrate experimentally several key aspects of this type of interferometer. We use spatially patterned light beams to excite counter-propagating sound waves within the condensate and use \emph{in situ} absorption imaging to characterize their evolution. We present an analysis technique by which we extract separately the oscillation frequencies of the standing-wave acoustic modes, the frequency splitting caused by static imperfections in the trapping potential, and the characteristic precession of the standing-wave pattern due to rotation. Supported by analytic and numerical calculations, we interpret the noise in our measurements, which is dominated by atom shot noise, in terms of rotation noise. While the noise of our acoustic interferometric sensor, at the level of $\sim \mbox{rad}\, \mbox{s}^{-1}/\sqrt{\mbox{Hz}}$, is high owing to rapid acoustic damping and the small radius of the trap, the proof-of-concept device does operate at $10^4 - 10^6$ times higher density and in a volume $10^9$ times smaller than free-falling atom interferometers., Comment: 11 pages including supplemental material, 7 figures

Spinor Bose gases form a family of quantum fluids manifesting both magnetic order and superfluidity. This article reviews experimental and theoretical progress in understanding the static and dynamic properties of these fluids. The connection between system properties and the rotational symmetry properties of the atomic states and their interactions are investigated. Following a review of the experimental techniques used for characterizing spinor gases, their mean-field and many-body ground states, both in isolation and under the application of symmetry-breaking external fields, are discussed. These states serve as the starting point for understanding low-energy dynamics, spin textures and topological defects, effects of magnetic dipole interactions, and various non-equilibrium collective spin-mixing phenomena. The paper aims to form connections and establish coherence among the vast range of works on spinor Bose gases, so as to point to open questions and future research opportunities., Comment: 60 page review paper, 24 figures. Revised following comments from colleagues and referees. We warmly invite further comments on the paper

The mechanical influence on objects due to their interaction with light has been a central topic in atomic physics for decades. Thus, one finds that many concepts developed to describe cavity optomechanical systems with solid-state mechanical oscillators have also been developed in parallel in relation to cold atomic physics. I describe several of these ideas from atomic physics, including optical methods for detecting quantum states of single cold atoms and atomic ensembles, motional effects within single-atom cavity quantum electrodynamics, and collective optical effects such as superradiant Rayleigh scattering and cavity cooling of atomic ensembles. I present several experimental realizations of cavity optomechanics in which an atomic ensemble serves as the mechanical element. These are divided between systems driven either by sending light onto the cavity input mirrors ("cavity pumped"), or by sending light onto the atomic ensemble ("side pumped"). The cavity-pumped systems clearly exhibit the key phenomena of cavity optomechanical systems, including cavity-aided position sensing, coherent back action effects such as the optical spring and cavity cooling, and optomechanical bistability; several of these effects have been detected not only for linear but also for quadratic optomechanical coupling. The extreme isolation of the atomic ensemble from mechanical disturbances, and its strong polarizability near the atomic resonance frequency, allow these optomechanical systems to be highly sensitive to quantum radiation pressure fluctuations. I describe several ways in which these fluctuations are observed experimentally. I conclude by considering the side-pumped cavity experiments in terms of cavity optomechanics, complementing recent treatments of these systems in terms of condensed-matter physics concepts such as quantum phase transitions and supersolidity., Comment: 43 page review paper. This manuscript will appear as a chapter in the forthcoming book entitled "Cavity optomechanics", edited by M. Aspelmeyer, T. Kippenberg, and F. Marquardt, to be published by Springer. Comments on this review are welcome

We directly measure the quantized collective motion of a gas of thousands of ultracold atoms, coupled to light in a high-finesse optical cavity. We detect strong asymmetries, as high as 3:1, in the intensity of light scattered into low- and high-energy motional sidebands. Owing to high cavity-atom cooperativity, the optical output of the cavity contains a spectroscopic record of the energy exchanged between light and motion, directly quantifying the heat deposited by a quantum position measurement's backaction. Such backaction selectively causes the phonon occupation of the observed collective modes to increase with the measurement rate. These results, in addition to providing a method for calibrating the motion of low-occupation mechanical systems, offer new possibilities for investigating collective modes of degenerate gases and for diagnosing optomechanical measurement backaction., Comment: 4 pages plus supplemental material

Geometrically frustrated systems with a large degeneracy of low energy states are of central interest in condensed-matter physics. The kagome net - a pattern of corner-sharing triangular plaquettes - presents a particularly high degree of frustration, reflected in the non-dispersive orbital bands. The ground state of the kagome quantum antiferromagnet, proposed to be a quantum spin liquid or valence bond solid, remains uncertain despite decades of work. Solid-state kagome magnets suffer from significant magnetic disorder or anisotropy that complicates the interpretation of experimental results. Here, we realize the kagome geometry in a two-dimensional optical superlattice for ultracold $^{87}$Rb atoms. We employ atom optics to characterize the lattice as it is tuned between various geometries, including kagome, one-dimensional stripe, and decorated triangular lattices, allowing for a sensitive control of frustration. The lattices implemented in this work offer a near-ideal realization of a paradigmatic model of many-body quantum physics., Comment: 5 pages, 4 figures, 1 table

Accessing distinctly quantum aspects of the interaction between light and the position of a mechanical object has been an outstanding challenge to cavity-optomechanical systems. Only cold-atom implementations of cavity optomechanics have indicated effects of the quantum fluctuations in the optical radiation pressure force. Here we use such a system, in which quantum photon-number fluctuations significantly drive the center of mass of an atomic ensemble inside a Fabry-Perot cavity. We show that the optomechanical response both amplifies and ponderomotively squeezes the quantum light field. We also demonstrate that classical optical fluctuations can be attenuated by 26 dB or amplified by 20 dB with a weak input pump power of < 40 pW, and characterize the optomechanical amplifier's frequency-dependent gain and phase response in both the amplitude and phase-modulation quadratures.