Spin squeezing of atomic ensembles via nuclear-electronic spin entanglement

Heisenberg-limited spin squeezing in a hybrid system with silicon-vacancy centers

In this paper, we investigate the spin squeezing in a hybrid quantum system consisting of a Silicon-Vacancy (SiV) center ensemble coupled to a diamond acoustic waveguide via the strain interaction. Two sets of non-overlapping driving fields, each contains two time-dependent microwave fields, are applied to this hybrid system. By modulating these fields, the one-axis twist (OAT) interaction and two-axis two-spin (TATS) interaction can be independently realized. In the latter case the squeezing parameter scales to spin number as ξ R2∼1.61N -0.64 with the consideration of dissipation, which is very close to the Heisenberg limit. Furthermore, this hybrid system allows for the study of spin squeezing generated by the simultaneous presence of OAT and TATS interactions, which reveals sensitivity to the parity of the number of spins Ntot, whether it is even or odd. Our scheme enriches the approach for generating Heisenberg-limited spin squeezing in spin-phonon hybrid systems and offers the possibility for future applications in quantum information processing.

Quantum sensors for biomedical applications

Quantum sensors are finding their way from laboratories to the real world, as witnessed by the increasing number of start-ups in this field. The atomic length scale of quantum sensors and their coherence properties enable unprecedented spatial resolution and sensitivity. Biomedical applications could benefit from these quantum technologies, but it is often difficult to evaluate the potential impact of the techniques. This Review sheds light on these questions, presenting the status of quantum sensing applications and discussing their path towards commercialization. The focus is on two promising quantum sensing platforms: optically pumped atomic magnetometers, and nitrogen-vacancy centres in diamond. The broad spectrum of biomedical applications is highlighted by four case studies ranging from brain imaging to single-cell spectroscopy.

Dipolar physics: a review of experiments with magnetic quantum gases

Since the achievement of quantum degeneracy in gases of chromium atoms in 2004, the experimental investigation of ultracold gases made of highly magnetic atoms has blossomed. The field has yielded the observation of many unprecedented phenomena, in particular those in which long-range and anisotropic dipole-dipole interactions (DDIs) play a crucial role. In this review, we aim to present the aspects of the magnetic quantum-gas platform that make it unique for exploring ultracold and quantum physics as well as to give a thorough overview of experimental achievements. Highly magnetic atoms distinguish themselves by the fact that their electronic ground-state configuration possesses a large electronic total angular momentum. This results in a large magnetic moment and a rich electronic transition spectrum. Such transitions are useful for cooling, trapping, and manipulating these atoms. The complex atomic structure and large dipolar moments of these atoms also lead to a dense spectrum of resonances in their two-body scattering behaviour. These resonances can be used to control the interatomic interactions and, in particular, the relative importance of contact over dipolar interactions. These features provide exquisite control knobs for exploring the few- and many-body physics of dipolar quantum gases. The study of dipolar effects in magnetic quantum gases has covered various few-body phenomena that are based on elastic and inelastic anisotropic scattering. Various many-body effects have also been demonstrated. These affect both the shape, stability, dynamics, and excitations of fully polarised repulsive Bose or Fermi gases. Beyond the mean-field instability, strong dipolar interactions competing with slightly weaker contact interactions between magnetic bosons yield new quantum-stabilised states, among which are self-bound droplets, droplet assemblies, and supersolids. Dipolar interactions also deeply affect the physics of atomic gases with an internal degree of freedom as these interactions intrinsically couple spin and atomic motion. Finally, long-range dipolar interactions can stabilise strongly correlated excited states of 1D gases and also impact the physics of lattice-confined systems, both at the spin-polarised level (Hubbard models with off-site interactions) and at the spinful level (XYZ models). In the present manuscript, we aim to provide an extensive overview of the various related experimental achievements up to the present.

Quantum enhancement of a single quantum battery by repeated interactions with large spins

A generalized collision model is developed to investigate coherent charging a single quantum battery by repeated interactions with many-atom large spins, where collective atom operators are adopted and the battery is modeled by a uniform energy ladder. For an initially empty battery, we derive analytical results of the average number of excitations and hence the charging power in the short-time limit. Our analytical results show that a faster charging and an increased amount of the power in the coherent protocol uniquely arise from the phase coherence of the atoms. Finally, we show that the charging power defined by the so-called ergotropy almost follows our analytical result, due to a nearly pure state of the battery in the short-time limit.

Metrological Characterization of Non-Gaussian Entangled States of Superconducting Qubits

Multipartite entangled states are significant resources for both quantum information processing and quantum metrology. In particular, non-Gaussian entangled states are predicted to achieve a higher sensitivity of precision measurements than Gaussian states. On the basis of metrological sensitivity, the conventional linear Ramsey squeezing parameter (RSP) efficiently characterizes the Gaussian entangled atomic states but fails for much wider classes of highly sensitive non-Gaussian states. These complex non-Gaussian entangled states can be classified by the nonlinear squeezing parameter (NLSP), as a generalization of the RSP with respect to nonlinear observables and identified via the Fisher information. However, the NLSP has never been measured experimentally. Using a 19-qubit programmable superconducting processor, we report the characterization of multiparticle entangled states generated during its nonlinear dynamics. First, selecting ten qubits, we measure the RSP and the NLSP by single-shot readouts of collective spin operators in several different directions. Then, by extracting the Fisher information of the time-evolved state of all 19 qubits, we observe a large metrological gain of 9.89_{-0.29}^{+0.28}  dB over the standard quantum limit, indicating a high level of multiparticle entanglement for quantum-enhanced phase sensitivity. Benefiting from high-fidelity full controls and addressable single-shot readouts, the superconducting processor with interconnected qubits provides an ideal platform for engineering and benchmarking non-Gaussian entangled states that are useful for quantum-enhanced metrology.

Minimal Model for Fast Scrambling

We study quantum information scrambling in spin models with both long-range all-to-all and short-range interactions. We argue that a simple global, spatially homogeneous interaction together with local chaotic dynamics is sufficient to give rise to fast scrambling, which describes the spread of quantum information over the entire system in a time that is logarithmic in the system size. This is illustrated in two tractable models: (1) a random circuit with Haar random local unitaries and a global interaction and (2) a classical model of globally coupled nonlinear oscillators. We use exact numerics to provide further evidence by studying the time evolution of an out-of-time-order correlator and entanglement entropy in spin chains of intermediate sizes. Our results pave the way towards experimental investigations of fast scrambling and aspects of quantum gravity with quantum simulators.

Maximal Quantum Fisher Information for Mixed States

We study quantum metrology for unitary dynamics. Analytic solutions are given for both the optimal unitary state preparation starting from an arbitrary mixed state and the corresponding optimal measurement precision. This represents a rigorous generalization of known results for optimal initial states and upper bounds on measurement precision which can only be saturated if pure states are available. In particular, we provide a generalization to mixed states of an upper bound on measurement precision for time-dependent Hamiltonians that can be saturated with optimal Hamiltonian control. These results make precise and reveal the full potential of mixed states for quantum metrology.

Quantum-enhanced sensing using non-classical spin states of a highly magnetic atom

Coherent superposition states of a mesoscopic quantum object play a major role in our understanding of the quantum to classical boundary, as well as in quantum-enhanced metrology and computing. However, their practical realization and manipulation remains challenging, requiring a high degree of control of the system and its coupling to the environment. Here, we use dysprosium atoms-the most magnetic element in its ground state-to realize coherent superpositions between electronic spin states of opposite orientation, with a mesoscopic spin size J = 8. We drive coherent spin states to quantum superpositions using non-linear light-spin interactions, observing a series of collapses and revivals of quantum coherence. These states feature highly non-classical behavior, with a sensitivity to magnetic fields enhanced by a factor 13.9(1.1) compared to coherent spin states-close to the Heisenberg limit 2J = 16-and an intrinsic fragility to environmental noise.

Squeezing giant spin states via geometric phase control in cavity-assisted Raman transitions

Squeezing ensemble of spins provides a way to surpass the standard quantum limit in quantum metrology and test the fundamental physics as well, and therefore attracts broad interest. Here we propose an experimentally accessible protocol to squeeze a giant ensemble of spins via the geometric phase control (GPC). Using the cavity-assisted Raman transition (CART) in a double Λ-type system, we realize an effective Dicke model. Under the condition of vanishing effective spin transition frequency, we find a particular evolution time where the cavity decouples from the spins and the spin ensemble is squeezed considerably. Our scheme combines the CART and the GPC, and has the potential to improve the sensitivity in quantum metrology with spins by about two orders.

Entanglement-Enhanced Radio-Frequency Field Detection and Waveform Sensing

We demonstrate a new technique for detecting the amplitude of arbitrarily chosen components of radio-frequency waveforms based on stroboscopic backaction evading measurements. We combine quantum nondemolition measurements and stroboscopic probing to detect waveform components with magnetic sensitivity beyond the standard quantum limit. Using an ensemble of 1.5×10^{6} cold rubidium atoms, we demonstrate entanglement-enhanced sensing of sinusoidal and linearly chirped waveforms, with 1.0(2) and 0.8(3) dB metrologically relevant noise reduction, respectively. We achieve volume-adjusted sensitivity of δBsqrt[V]≈3.96  fTsqrt[cm^{3}/Hz], comparable to the best rf magnetometers.

Lifetime assessment of RbN3-filled MEMS atomic vapor cells with Al2O3 coating

Micro-fabricated (MEMS) alkali vapor cells are at the heart of the miniaturization of atomic devices such as atomic magnetometers, atomic gyroscopes and atomic clocks. Among the different techniques used to fill microfabricated alkali vapor cell, UV decomposition of rubidium azide (RbN₃) into metallic Rb and nitrogen in Al₂O₃ coated cells is a very promising approach for low-cost wafer-level fabrication. Here we present a detailed lifetime study of such cells. The rubidium consumption being the main identified cell failure mode, it is monitored with an novel image analysis technique and with high temperature long term aging tests.

Revealing nonclassicality beyond Gaussian states via a single marginal distribution

A standard method to obtain information on a quantum state is to measure marginal distributions along many different axes in phase space, which forms a basis of quantum-state tomography. We theoretically propose and experimentally demonstrate a general framework to manifest nonclassicality by observing a single marginal distribution only, which provides a unique insight into nonclassicality and a practical applicability to various quantum systems. Our approach maps the 1D marginal distribution into a factorized 2D distribution by multiplying the measured distribution or the vacuum-state distribution along an orthogonal axis. The resulting fictitious Wigner function becomes unphysical only for a nonclassical state; thus the negativity of the corresponding density operator provides evidence of nonclassicality. Furthermore, the negativity measured this way yields a lower bound for entanglement potential-a measure of entanglement generated using a nonclassical state with a beam-splitter setting that is a prototypical model to produce continuous-variable (CV) entangled states. Our approach detects both Gaussian and non-Gaussian nonclassical states in a reliable and efficient manner. Remarkably, it works regardless of measurement axis for all non-Gaussian states in finite-dimensional Fock space of any size, also extending to infinite-dimensional states of experimental relevance for CV quantum informatics. We experimentally illustrate the power of our criterion for motional states of a trapped ion, confirming their nonclassicality in a measurement-axis-independent manner. We also address an extension of our approach combined with phase-shift operations, which leads to a stronger test of nonclassicality, that is, detection of genuine non-Gaussianity under a CV measurement.

Quantum communication with macroscopically bright nonclassical states

We analyze homodyne detection of macroscopically bright multimode nonclassical states of light and propose their application in quantum communication. We observe that the homodyne detection is sensitive to a mode-matching of the bright light to the highly intense local oscillator. Unmatched bright modes of light result in additional noise which technically limits detection of Gaussian entanglement at macroscopic level. When the mode-matching is sufficient, we show that multimode quantum key distribution with bright beams is feasible. It finally merges the quantum communication with classical optical technology of visible beams of light.

Quantum entanglement at ambient conditions in a macroscopic solid-state spin ensemble

Entanglement is a key resource for quantum computers, quantum-communication networks, and high-precision sensors. Macroscopic spin ensembles have been historically important in the development of quantum algorithms for these prospective technologies and remain strong candidates for implementing them today. This strength derives from their long-lived quantum coherence, strong signal, and ability to couple collectively to external degrees of freedom. Nonetheless, preparing ensembles of genuinely entangled spin states has required high magnetic fields and cryogenic temperatures or photochemical reactions. We demonstrate that entanglement can be realized in solid-state spin ensembles at ambient conditions. We use hybrid registers comprising of electron-nuclear spin pairs that are localized at color-center defects in a commercial SiC wafer. We optically initialize 10(3) identical registers in a 40-μm(3) volume (with [Formula: see text] fidelity) and deterministically prepare them into the maximally entangled Bell states (with 0.88 ± 0.07 fidelity). To verify entanglement, we develop a register-specific quantum-state tomography protocol. The entanglement of a macroscopic solid-state spin ensemble at ambient conditions represents an important step toward practical quantum technology.

Alkali-Metal Spin Maser

Quantum measurement is a combination of a read-out and a perturbation of the quantum system. We explore the nonlinear spin dynamics generated by a linearly polarized probe beam in a continuous measurement of the collective spin state in a thermal alkali-metal atomic sample. We demonstrate that the probe-beam-driven perturbation leads, in the presence of indirect pumping, to complete polarization of the sample and macroscopic coherent spin oscillations. As a consequence of the former we report observation of spectral profiles free from collisional broadening. Nonlinear dynamics is studied through exploring its effect on radio frequency as well as spin noise spectra.

Spin squeezing in a quadrupolar nuclei NMR system

We have produced and characterized spin-squeezed states at a temperature of 26 °C in a nuclear magnetic resonance quadrupolar system. The experiment was carried out on 133Cs nuclei of spin I=7/2 in a sample of lyotropic liquid crystal. The source of spin squeezing was identified as the interaction between the quadrupole moment of the nuclei and the electric field gradients present within the molecules. We use the spin angular momentum representation to describe formally the nonlinear operators that produce the spin squeezing on a Hilbert space of dimension 2I+1=8. The quantitative and qualitative characterization of this spin-squeezing phenomenon is expressed by a squeezing parameter and squeezing angle developed for the two-mode Bose-Einstein condensate system, as well as by the Wigner quasiprobability distribution function. The generality of the present experimental scheme points to potential applications in solid-state physics.

Magnetic sensitivity beyond the projection noise limit by spin squeezing

We report the generation of spin squeezing and entanglement in a magnetically sensitive atomic ensemble, and entanglement-enhanced field measurements with this system. A maximal m(f) = ± 1 Raman coherence is prepared in an ensemble of 8.5 × 10(5) laser-cooled (87)Rb atoms in the f = 1 hyperfine ground state, and the collective spin is squeezed by synthesized optical quantum nondemolition measurement. This prepares a state with large spin alignment and noise below the projection-noise level in a mixed alignment-orientation variable. 3.2 dB of noise reduction is observed and 2.0 dB of squeezing by the Wineland criterion, implying both entanglement and metrological advantage. Enhanced sensitivity is demonstrated in field measurements using alignment-to-orientation conversion.

Enhanced squeezing of a collective spin via control of its qudit subsystems

Unitary control of qudits can improve the collective spin squeezing of an atomic ensemble. Preparing the atoms in a state with large quantum fluctuations in magnetization strengthens the entangling Faraday interaction. The resulting increase in interatomic entanglement can be converted into metrologically useful spin squeezing. Further control can squeeze the internal atomic spin without compromising entanglement, providing an overall multiplicative factor in the collective squeezing. We model the effects of optical pumping and study the tradeoffs between enhanced entanglement and decoherence. For realistic parameters we see improvements of ~10 dB.

Collective state measurement of mesoscopic ensembles with single-atom resolution

We demonstrate single-atom resolution, as well as detection sensitivity more than 20 dB below the quantum projection noise limit, for hyperfine-state-selective measurements on mesoscopic ensembles containing 100 or more atoms. The measurement detects the atom-induced shift of the resonance frequency of an optical cavity containing the ensemble. While spatially varying coupling of atoms to the cavity prevents the direct observation of a quantized signal, the demonstrated measurement resolution provides the readout capability necessary for atomic interferometry substantially below the standard quantum limit and down to the Heisenberg limit.

Tomography of a high-purity narrowband photon from a transient atomic collective excitation

We demonstrate efficient heralded generation of high purity narrow-bandwidth single photons from a transient collective spin excitation in a hot atomic vapor cell. Employing optical homodyne tomography, we fully reconstruct the density matrix of the generated photon and observe a Wigner function reaching the zero value without correcting for any inefficiencies. The narrow bandwidth of the photon produced is accompanied by a high generation rate yielding a high spectral brightness. The source is, therefore, compatible with atomic-based quantum memories as well as other applications in light-atom interfacing. This Letter paves the way to preparing and measuring arbitrary superposition states of collective atomic excitations.

Breakdown of the classical description of a local system

We provide a straightforward demonstration of a fundamental difference between classical and quantum mechanics for a single local system: namely, the absence of a joint probability distribution of the position x and momentum p. Elaborating on a recently reported criterion by Bednorz and Belzig [Phys. Rev. A 83, 052113 (2011)] we derive a simple criterion that must be fulfilled for any joint probability distribution in classical physics. We demonstrate the violation of this criterion using the homodyne measurement of a single photon state, thus proving a straightforward signature of the breakdown of a classical description of the underlying state. Most importantly, the criterion used does not rely on quantum mechanics and can thus be used to demonstrate nonclassicality of systems not immediately apparent to exhibit quantum behavior. The criterion is directly applicable to any system described by the continuous canonical variables x and p, such as a mechanical or an electrical oscillator and a collective spin of a large ensemble.

A low phase noise microwave source for atomic spin squeezing experiments in 87Rb

We describe and characterize a simple, low cost, low phase noise microwave source that operates near 6.800 GHz for agile, coherent manipulation of ensembles of (87)Rb. Low phase noise is achieved by directly multiplying a low phase noise 100 MHz crystal to 6.8 GHz using a nonlinear transmission line and filtering the output with custom band-pass filters. The fixed frequency signal is single sideband modulated with a direct digital synthesis frequency source to provide the desired phase, amplitude, and frequency control. Before modulation, the source has a single sideband phase noise near -140 dBc/Hz in the range of 10 kHz-1 MHz offset from the carrier frequency and -130 dBc/Hz after modulation. The resulting source is estimated to contribute added spin-noise variance 16 dB below the quantum projection noise level during quantum nondemolition measurements of the clock transition in an ensemble 7 × 10(5) (87)Rb atoms.

Spin squeezing inequalities for arbitrary spin

We determine the complete set of generalized spin squeezing inequalities, given in terms of the collective angular momentum components, for particles with an arbitrary spin. They can be used for the experimental detection of entanglement in an ensemble in which the particles cannot be individually addressed. We also present a large set of criteria involving collective observables different from the angular momentum coordinates. We show that some of the inequalities can be used to detect k-particle entanglement and bound entanglement.

Generating entanglement and squeezed states of nuclear spins in quantum dots

We present a scheme for achieving coherent spin squeezing of nuclear spin states in semiconductor quantum dots. The nuclear polarization dependence of the electron spin resonance generates a unitary evolution that drives nuclear spins into a collective entangled state. The polarization dependence of the resonance generates an area-preserving, twisting dynamics that squeezes and stretches the nuclear spin Wigner distribution without the need for nuclear spin flips. Our estimates of squeezing times indicate that the entanglement threshold can be reached in current experiments.

Spin squeezing: transforming one-axis twisting into two-axis twisting

Squeezed spin states possess unique quantum correlation or entanglement and are significantly promising for advancing quantum information processing and quantum metrology. In recent back-to-back publications [C. Gross et al., Nature (London) 464, 1165 (2010) and Max F. Riedel et al., Nature (London) 464, 1170 (2010)], reduced spin fluctuations are observed leading to spin squeezing at -8.2 and -2.5 dB, respectively, in two-component atomic condensates exhibiting one-axis-twisting interactions. The noise reduction limit for the one-axis twisting scales as ∝1/N(2/3), which for a condensate with N∼10(3) atoms is about 100 times below the standard quantum limit. We present a scheme using repeated Rabi pulses capable of transforming the one-axis-twisting spin squeezing into the two-axis-twisting type, leading to Heisenberg limited noise reduction ∝1/N or an extra tenfold improvement for N∼10(3).

Directly estimating nonclassicality

We establish a method of directly measuring and estimating nonclassicality--operationally defined in terms of the distinguishability of a given state from one with a positive Wigner function. It allows us to certify nonclassicality, based on possibly much fewer measurement settings than necessary for obtaining complete tomographic knowledge, and is at the same time equipped with a full certificate. We find that even from measuring two conjugate variables alone, one may infer the nonclassicality of quantum mechanical modes. This method also provides a practical tool to eventually certify such features in mechanical degrees of freedom in opto-mechanics. The proof of the result is based on Bochner's theorem characterizing classical and quantum characteristic functions and on semidefinite programming. In this joint theoretical-experimental work we present data from experimental optical Fock state preparation.

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. 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 interest. 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. Extending quantum interferometry 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' scheme 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 atoms.

Mesoscopic atomic entanglement for precision measurements beyond the standard quantum limit

Squeezing of quantum fluctuations by means of entanglement is a well-recognized goal in the field of quantum information science and precision measurements. In particular, squeezing the fluctuations via entanglement between 2-level atoms can improve the precision of sensing, clocks, metrology, and spectroscopy. Here, we demonstrate 3.4 dB of metrologically relevant squeezing and entanglement for greater, similar 10(5) cold caesium atoms via a quantum nondemolition (QND) measurement on the atom clock levels. We show that there is an optimal degree of decoherence induced by the quantum measurement which maximizes the generated entanglement. A 2-color QND scheme used in this paper is shown to have a number of advantages for entanglement generation as compared with a single-color QND measurement.

Entanglement, nonlinear dynamics, and the heisenberg limit

We show that quantum Fisher information provides a sufficient condition to recognize multiparticle entanglement in an N qubit state. The same criterion gives a necessary and sufficient condition for sub-shot-noise phase sensitivity in the estimation of a collective rotation angle theta. The analysis therefore singles out the class of entangled states which are useful to overcome classical phase sensitivity in metrology and sensors. We finally study the creation of useful entangled states by the nonlinear dynamical evolution of two decoupled Bose-Einstein condensates or trapped ions.