Using squeezed states it is possible to surpass the standard quantum limit of measurement uncertainty by reducing the measurement uncertainty of one property at the expense of another complementary property [1]. Squeezed states were first demonstrated in optical fields [2] and later with ensembles of pseudo spin-1/2 atoms using non-linear atom-light interactions [3]. Recently, collisional interactions in ultracold atomic gases have been used to generate a large degree of quadrature spin squeezing in two-component Bose condensates [4,5]. For pseudo spin-1/2 systems, the complementary properties are the different components of the total spin vector S , which fully characterize the state on an SU(2) Bloch sphere. Here, we measure squeezing in a spin-1 Bose condensate, an SU(3) system, which requires measurement of the rank-2 nematic or quadrupole tensor Q ij as well to fully characterize the state. Following a quench through a nematic to ferromagnetic quantum phase transition, squeezing is observed in the variance of the quadratures up to −8.3 +0.6 −0.7 dB (−10.3 +0.7 −0.9 dB corrected for detection noise) below the standard quantum limit. This spin-nematic squeezing is observed for negligible occupation of the squeezed modes and is analogous to optical two-mode vacuum squeezing. This work has potential applications to continuous variable quantum information and quantum-enhanced magnetometry.The study of many-body quantum entangled states including atomic spin squeezed states is an active research frontier. In addition to being intrinsically fascinating, such states have applications in precision measurements [6], quantum information and fundamental tests of quantum mechanics [7]. Atomic squeezed states were first considered for ensembles of two-level (pseudo spin-1/2) atoms. For spin-1/2 particles, coherent states of the system are uniquely specified by the components of the total spin vector S , typically illustrated on a SU(2) Bloch sphere. For particles with higher spin, additional degrees of freedom beyond the spin vector are required to fully specify the state. For spin-1 particles, a natural basis to describe the wavefunction is the SU(3) Cartesian dipole-quadrupole basis, consisting of the three components of the spin vector,Ŝ i , and the moments of the rank-2 quadrupole or nematic tensor,Q ij ({i, j} ∈ {x, y, z}). In matrix form, the nematic moments can be written. Spin-1 atomic Bose-Einstein condensates [9-13] provide an exceptionally clean experimental platform to investigate the quantum dynamics of many-body spin sys-tems. They feature controllable quantum phase transitions, well-understood underlying microscopic models, and flexible defect-free geometries. Importantly, it is possible to initialize non-equilibrium or excited states of the system and to directly measure both the spin vector and the nematic tensor using standard atomic state manipulation tools. Law, et al., demonstrated that the spinor interaction can be written as total spin angular momentum, λŜ 2 whereŜ 2 =Ŝ 2x +Ŝ 2 y +Ŝ 2 z [14]. It is ...

Levitated optomechanics has great potential in precision measurements, thermodynamics, macroscopic quantum mechanics, and quantum sensing. Here we synthesize and optically levitate silica nanodumbbells in high vacuum. With a linearly polarized laser, we observe the torsional vibration of an optically levitated nanodumbbell. This levitated nanodumbbell torsion balance is a novel analog of the Cavendish torsion balance, and provides rare opportunities to observe the Casimir torque and probe the quantum nature of gravity as proposed recently. With a circularly polarized laser, we drive a 170-nm-diameter nanodumbbell to rotate beyond 1 GHz, which is the fastest nanomechanical rotor realized to date. Smaller silica nanodumbbells can sustain higher rotation frequencies. Such ultrafast rotation may be used to study material properties and probe vacuum friction.

An optically levitated nanoparticle in vacuum is a paradigm optomechanical system for sensing and studying macroscopic quantum mechanics. While its center-of-mass motion has been investigated intensively, its torsional vibration has only been studied theoretically in limited cases. Here we report the first experimental observation of the torsional vibration of an optically levitated nonspherical nanoparticle in vacuum. We achieve this by utilizing the coupling between the spin angular momentum of photons and the torsional vibration of a nonspherical nanoparticle whose polarizability is a tensor. The torsional vibration frequency can be one order of magnitude higher than its center-of-mass motion frequency, which is promising for ground state cooling. We propose a simple yet novel scheme to achieve ground state cooling of its torsional vibration with a linearly-polarized Gaussian cavity mode. A levitated nonspherical nanoparticle in vacuum will also be an ultrasensitive nanoscale torsion balance with a torque detection sensitivity on the order of 10 −29 N · m/ √ Hz under realistic conditions. An optically levitated dielectric particle in vacuum [1][2][3] is an ultrasensitive detector for force sensing [4,5], millicharge searching [6] and other applications [7,8]. It will provide a great platform to test fundamental theories such as objective collapse models [9, 10] and quantum gravity [11] when its mechanical motion can be cooled to the quantum regime [12,13]. Recently, feedback cooling of the center-of-mass (COM) motion of a levitated nanosphere to about 450 µK (about 63 phonons at 150 kHz) [14], and cavity cooling of the COM motion of a nanosphere to a few mK [15] were demonstrated. The vibration mode would have already been in ground state at 450 µK [14] if its frequency is above 10 MHz. Increasing the vibration frequency of the nanoparticle can be a key to achieve ground state cooling. However, this can not be achieved by simply increasing the intensity of the trapping laser, which induces heating and subsequently causes the loss of the nanoparticle [4,16]. Besides COM motion, a pioneering work has proposed to use multiple Laguerre-Gaussian (LG) cavity modes to achieve angular trapping of a dielectric rod and cool its torsional vibration (TOR) to the ground state [12]. This was later generalized to micro-windmills [17], which have better overlap with LG cavity modes. These intriguing proposals of torsional optomechanics, however, have not been realized experimentally yet.In this work, we report the first experimental observation of the torsional vibration of an optically levitated nonspherical nanoparticle in vacuum, and show that the torsional frequency can be one order of magnitude higher than the COM frequency at the same laser intensity. We explain our observation using a model of an ellipsoidal nanoparticle levitated by a linearly-polarized Gaussian beam. For an ellipsoid much smaller than the wavelength of the trapping laser, its polarizability is a tensor due to its geometry [18]. In a linearly polariz...

We observe power-law scaling of the temporal onset of excitations with quench speed in the neighborhood of the quantum phase transition between the polar and broken-axisymmetry phases in a small spin-1 ferromagnetic Bose-Einstein condensate. As the system is driven through the quantum critical point by tuning the Hamiltonian, the vanishing energy gap between the ground state and first excited state causes the reaction time scale of the system to diverge, preventing it from adiabatically following the ground state. We measure the temporal evolution of the spin populations for different quench speeds and determine the exponents characterizing the scaling of the onset of excitations, which are in good agreement with the predictions of the Kibble-Zurek mechanism.In a second-order (or continuous) quantum phase transition (QPT), a qualitative change in the system's ground state occurs at zero temperature when a parameter in the Hamiltonian is varied across a quantum critical point (QCP) [1]. Near the critical point of the transition, the time scale characterizing the dynamics of a system diverges, and the scaling of this divergence with respect to the quench speed through the phase transition is characterized by universal critical exponents. The Kibble-Zurek mechanism (KZM) as originally formulated characterizes the formation of topological defects when a system undergoes a continuous phase transition at a finite rate. This concept was first conceived by Kibble in his study on topology of cosmic domains and strings in the early universe [2,3], and it was later extended by Zurek [4][5][6] who suggested applying these symmetry breaking ideas to condensed matter systems, such as superconductors and superfluids. This seminal work was followed by many theoretical studies applying the KZM to cosmology, condensed matter, cold atoms and more [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. In parallel, the KZM has been studied experimentally and verified in a large variety of systems, including liquid crystals [25,26] [33]. There has also been significant interest in the KZM in the cold atom community. In recent years, it has been observed in ion chains [34][35][36][37], in atomic gases in optical lattices [38], and in Bose-Einstein condensates (BECs), through the formation of spatial domains during condensation [39,40], vortices [41,42], creation of solitons [43] and supercurrents [44]. Only a few experiments have explored the KZM using QPTs (i.e. at zero temperature), namely an investigation of the Mott insulator to superfluid transition [45] and, in a recent preprint [46], an ion chain cooled to the ground state. There has been related work investigating universal scaling in optical lattices [47] and recently in the miscible-immiscible transition in a two-component Bose gas [48].A ferromagnetic spin-1 ( 87 Rb) BEC exhibits a QPT between a symmetric polar (P) phase and a brokenaxisymmetry (BA) phase [20,49] due to the competition between magnetic and collisional spin interaction energies. There have be...

Electron spins of diamond nitrogen-vacancy (NV) centres are important quantum resources for nanoscale sensing and quantum information. Combining NV spins with levitated optomechanical resonators will provide a hybrid quantum system for novel applications. Here we optically levitate a nanodiamond and demonstrate electron spin control of its built-in NV centres in low vacuum. We observe that the strength of electron spin resonance (ESR) is enhanced when the air pressure is reduced. To better understand this system, we investigate the effects of trap power and measure the absolute internal temperature of levitated nanodiamonds with ESR after calibration of the strain effect. We also observe that oxygen and helium gases have different effects on both the photoluminescence and the ESR contrast of nanodiamond NV centres, indicating potential applications of NV centres in oxygen gas sensing. Our results pave the way towards a levitated spin–optomechanical system for studying macroscopic quantum mechanics.

A pendulum prepared perfectly inverted and motionless is a prototype of unstable equilibrium and corresponds to an unstable hyperbolic fixed point in the dynamical phase space. Here, we measure the non-equilibrium dynamics of a spin-1 Bose-Einstein condensate initialized as a minimum uncertainty spin-nematic state to a hyperbolic fixed point of the phase space. Quantum fluctuations lead to non-linear spin evolution along a separatrix and non-Gaussian probability distributions that are measured to be in good agreement with exact quantum calculations up to 0.25 s. At longer times, atomic loss due to the finite lifetime of the condensate leads to larger spin oscillation amplitudes, as orbits depart from the separatrix. This demonstrates how decoherence of a many-body system can result in apparent coherent behaviour. This experiment provides new avenues for studying macroscopic spin systems in the quantum limit and for investigations of important topics in non-equilibrium quantum dynamics.

Hybrid spin-mechanical systems have great potentials in sensing, macroscopic quantum mechanics, and quantum information science. In order to induce strong coupling between an electron spin and the center-of-mass motion of a mechanical oscillator, a large magnetic gradient is usually required, which is difficult to achieve. Here we show that strong coupling between the electron spin of a nitrogen-vacancy (NV) center and the torsional vibration of an optically levitated nanodiamond can be achieved in a uniform magnetic field. Thanks to the uniform magnetic field, multiple spins can strongly couple to the torsional vibration at the same time. We propose to utilize this new coupling mechanism to realize the Lipkin-Meshkov-Glick (LMG) model by an ensemble of NV centers in a levitated nanodiamond. The quantum phase transition in the LMG model and finite number effects can be observed with this system. We also propose to generate torsional superposition states and realize torsional matter-wave interferometry with spin-torsional coupling.

Nonequilibrium processes of small systems such as molecular machines are ubiquitous in biology, chemistry, and physics but are often challenging to comprehend. In the past two decades, several exact thermodynamic relations of nonequilibrium processes, collectively known as fluctuation theorems, have been discovered and provided critical insights. These fluctuation theorems are generalizations of the second law and can be unified by a differential fluctuation theorem. Here we perform the first experimental test of the differential fluctuation theorem using an optically levitated nanosphere in both underdamped and overdamped regimes and in both spatial and velocity spaces. We also test several theorems that can be obtained from it directly, including a generalized Jarzynski equality that is valid for arbitrary initial states, and the Hummer-Szabo relation. Our study experimentally verifies these fundamental theorems and initiates the experimental study of stochastic energetics with the instantaneous velocity measurement.