Deterministic Squeezed States with Collective Measurements and Feedback

Momentum-exchange interactions in a Bragg atom interferometer suppress Doppler dephasing

Large ensembles of laser-cooled atoms interacting through infinite-range photon-mediated interactions are powerful platforms for quantum simulation and sensing. Here we realize momentum-exchange interactions in which pairs of atoms exchange their momentum states by collective emission and absorption of photons from a common cavity mode, a process equivalent to a spin-exchange or XX collective Heisenberg interaction. The momentum-exchange interaction leads to an observed all-to-all Ising-like interaction in a matter-wave interferometer. A many-body energy gap also emerges, effectively binding interferometer matter-wave packets together to suppress Doppler dephasing in analogy to Mössbauer spectroscopy. The tunable momentum-exchange interaction expands the capabilities of quantum interaction-enhanced matter-wave interferometry and may enable the realization of exotic behaviors, including simulations of superconductors and dynamical gauge fields.

Universal Control of Symmetric States Using Spin Squeezing

The manipulation of quantum many-body systems is a crucial goal in quantum science. Entangled quantum states that are symmetric under qubits permutation are of growing interest. Yet, the creation and control of symmetric states has remained a challenge. Here, we introduce a method to universally control symmetric states, proposing a scheme that relies solely on coherent rotations and spin squeezing. We present protocols for the creation of different symmetric states including Schrödinger's cat and Gottesman-Kitaev-Preskill states. The obtained symmetric states can be transferred to traveling photonic states via spontaneous emission, providing a powerful approach for engineering desired quantum photonic states.

Dicke State Generation and Extreme Spin Squeezing via Rapid Adiabatic Passage

Considering the unique energy level structure of the one-axis twisting Hamiltonian in combination with standard rotations, we propose the implementation of a rapid adiabatic passage scheme on the Dicke state basis. The method permits to drive Dicke states of the many-atom system into entangled states with maximum quantum Fisher information. The designed states allow us to overcome the classical limit of phase sensitivity in quantum metrology and sensing. We show how to generate superpositions of Dicke states, which maximize metrological gain for a Ramsey interferometric measurement. The proposed scheme is remarkably robust to variations of the driving field and has favorable time scaling, especially for a small to moderate (∼1000) number of atoms, where the total time does not depend on the number of atoms.

Squeezing Multilevel Atoms in Dark States via Cavity Superradiance

We describe a method to create and store scalable and long-lived entangled spin-squeezed states within a manifold of many-body cavity dark states using collective emission of light from multilevel atoms inside an optical cavity. We show that the system can be tuned to generate squeezing in a dark state where it will be immune to superradiance. We also show more generically that squeezing can be generated using a combination of superradiance and coherent driving in a bright state, and subsequently be transferred via single-particle rotations to a dark state where squeezing can be stored. Our findings, readily testable in current optical cavity experiments with alkaline-earth-like atoms, can open a path for dissipative generation and storage of metrologically useful states in optical transitions.

Quantum Fluctuation Dynamics of Dispersive Superradiant Pulses in a Hybrid Light-Matter System

We consider theoretically a driven-dissipative quantum many-body system consisting of an atomic ensemble in a single-mode optical cavity as described by the open Tavis-Cummings model. In this hybrid light-matter system, the interplay between coherent and dissipative processes leads to superradiant pulses with a buildup of strong correlations, even for systems comprising hundreds to thousands of particles. A central feature of the mean-field dynamics is a self-reversal of two spin degrees of freedom due to an underlying time-reversal symmetry, which is broken by quantum fluctuations. We demonstrate a quench protocol that can maintain highly non-Gaussian states over long timescales. This general mechanism offers interesting possibilities for the generation and control of complex fluctuation patterns, as suggested for the improvement of quantum sensing protocols for dissipative spin amplification.

Quantum-enhanced sensing on optical transitions through finite-range interactions

The control over quantum states in atomic systems has led to the most precise optical atomic clocks so far1-3. Their sensitivity is bounded at present by the standard quantum limit, a fundamental floor set by quantum mechanics for uncorrelated particles, which can-nevertheless-be overcome when operated with entangled particles. Yet demonstrating a quantum advantage in real-world sensors is extremely challenging. Here we illustrate a pathway for harnessing large-scale entanglement in an optical transition using 1D chains of up to 51 ions with interactions that decay as a power-law function of the ion separation. We show that our sensor can emulate many features of the one-axis-twisting (OAT) model, an iconic, fully connected model known to generate scalable squeezing4 and Greenberger-Horne-Zeilinger-like states5-8. The collective nature of the state manifests itself in the preservation of the total transverse magnetization, the reduced growth of the structure factor, that is, spin-wave excitations (SWE), at finite momenta, the generation of spin squeezing comparable with OAT (a Wineland parameter9,10 of -3.9 ± 0.3 dB for only N = 12 ions) and the development of non-Gaussian states in the form of multi-headed cat states in the Q-distribution. We demonstrate the metrological utility of the states in a Ramsey-type interferometer, in which we reduce the measurement uncertainty by -3.2 ± 0.5 dB below the standard quantum limit for N = 51 ions.

Spin Squeezing by Rydberg Dressing in an Array of Atomic Ensembles

We report on the creation of an array of spin-squeezed ensembles of cesium atoms via Rydberg dressing, a technique that offers optical control over local interactions between neutral atoms. We optimize the coherence of the interactions by a stroboscopic dressing sequence that suppresses super-Poissonian loss. We thereby prepare squeezed states of N=200 atoms with a metrological squeezing parameter ξ^{2}=0.77(9) quantifying the reduction in phase variance below the standard quantum limit. We realize metrological gain across three spatially separated ensembles in parallel, with the strength of squeezing controlled by the local intensity of the dressing light. Our method can be applied to enhance the precision of tests of fundamental physics based on arrays of atomic clocks and to enable quantum-enhanced imaging of electromagnetic fields.

PT-Symmetric Feedback Induced Linewidth Narrowing

Narrow linewidth is a long-pursued goal in precision measurement and sensing. We propose a parity-time symmetric (PT-symmetric) feedback method to narrow the linewidths of resonance systems. By using a quadrature measurement-feedback loop, we transform a dissipative resonance system into a PT-symmetric system. Unlike the conventional PT-symmetric systems that typically require two or more modes, here the PT-symmetric feedback system contains only a single resonance mode, which greatly extends the scope of applications. The method enables remarkable linewidth narrowing and enhancement of measurement sensitivity. We illustrate the concept in a thermal ensemble of atoms, achieving a 48-fold narrowing of the magnetic resonance linewidth. By applying the method in magnetometry, we realize a 22-times improvement of the measurement sensitivity. This work opens the avenue for studying non-Hermitian physics and high-precision measurements in resonance systems with feedback.

Enhancing the force sensitivity of a squeezed light optomechanical interferometer

Application of frequency-dependent squeezed vacuum improves the force sensitivity of an optomechanical interferometer beyond the standard quantum limit by a factor of e^-r, where r is the squeezing parameter. In this work, we show that the application of squeezed light along with quantum back-action nullifying meter in an optomechanical cavity with mechanical mirror in middle configuration can enhance the sensitivity beyond the standard quantum limit by a factor of e^-reff, where r_eff = r + ln(4Δ/ζ)/2, for 0 < ζ/Δ < 1, with ζ as the optomechanical cavity decay rate and Δ as the detuning between cavity eigenfrequency and driving field. The technique described in this work is restricted to frequencies much smaller than the resonance frequency of the mechanical mirror. We further studied the sensitivity as a function of temperature, mechanical mirror reflectivity, and input laser power.

Weak-measurement-induced heating in Bose-Einstein condensates

Ultracold atoms are an ideal platform for understanding system-reservoir dynamics of many-body systems. Here, we study quantum back-action in atomic Bose-Einstein condensates, weakly interacting with a far-from resonant, i.e., dispersively interacting, probe laser beam. The light scattered by the atoms can be considered as a part of quantum measurement process, whereby the change in the system state derives from measurement back-action. We experimentally quantify the resulting back-action in terms of the deposited energy. We model the interaction of the system and environment with a generalized measurement process, leading to a Markovian reservoir. Further, we identify two systematic sources of heating and loss: a stray optical lattice and probe-induced light-assisted collisions (an intrinsic atomic process). The observed heating and loss rates are larger for blue detuning than for red detuning, where they are oscillatory functions of detuning with increased loss at molecular resonances and reduced loss between molecular resonances.

Entanglement-enhanced matter-wave interferometry in a high-finesse cavity

An ensemble of atoms can operate as a quantum sensor by placing atoms in a superposition of two different states. Upon measurement of the sensor, each atom is individually projected into one of the two states. Creating quantum correlations between the atoms, that is entangling them, could lead to resolutions surpassing the standard quantum limit^1–3 set by projections of individual atoms. Large amounts of entanglement^4–6 involving the internal degrees of freedom of laser-cooled atomic ensembles^4–16 have been generated in collective cavity quantum-electrodynamics systems, in which many atoms simultaneously interact with a single optical cavity mode. Here we report a matter-wave interferometer in a cavity quantum-electrodynamics system of 700 atoms that are entangled in their external degrees of freedom. In our system, each individual atom falls freely under gravity and simultaneously traverses two paths through space while entangled with the other atoms. We demonstrate both quantum non-demolition measurements and cavity-mediated spin interactions for generating squeezed momentum states with directly observed sensitivity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\,.\,{4}_{-0.9}^{+1.1}$$\end{document}3.4−0.9+1.1 dB and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\,.\,{5}_{-0.6}^{+0.6}$$\end{document}2.5−0.6+0.6 dB below the standard quantum limit, respectively. We successfully inject an entangled state into a Mach–Zehnder light-pulse interferometer with directly observed sensitivity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\,.\,{7}_{-0.5}^{+0.5}$$\end{document}1.7−0.5+0.5 dB below the standard quantum limit. The combination of particle delocalization and entanglement in our approach may influence developments of enhanced inertial sensors^17,18, searches for new physics, particles and fields^19–23, future advanced gravitational wave detectors^24,25 and accessing beyond mean-field quantum many-body physics^26–30.

A matter-wave interferometer is demonstrated with an interferometric phase noise below the standard quantum limit, combining two core concepts of quantum mechanics, that a particle can simultaneously be in two places at once and entanglement between distinct particles.

Measurement-based preparation of multimode mechanical states

Nanomechanical resonators are a key tool for future quantum technologies, such as quantum force sensors and interfaces, and for studies of macroscopic quantum physics. The ability to prepare room temperature nonclassical states is a major outstanding challenge. It has been suggested that this could be achieved using a fast continuous measurement to break the usual symmetry between position and momentum. Here, we demonstrate this symmetry breaking and use it to prepare a thermally squeezed mechanical state. Our experiments take advantage of collective measurements on multiple mechanical modes, which we show can increase the measurement speed and improve state preparation. Theoretically, we show that this result extends to the quantum regime, relaxing the requirements to generate nonclassical states. We predict that multimode conditioning can enable room temperature quantum squeezing with existing technology. Our work paves the way toward room temperature quantum nanomechanical devices and toward their application in quantum technology and fundamental science.

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.

Quantum Enhanced Cavity QED Interferometer with Partially Delocalized Atoms in Lattices

We propose a quantum enhanced interferometric protocol for gravimetry and force sensing using cold atoms in an optical lattice supported by a standing-wave cavity. By loading the atoms in partially delocalized Wannier-Stark states, it is possible to cancel the undesirable inhomogeneities arising from the mismatch between the lattice and cavity fields and to generate spin squeezed states via a uniform one-axis twisting model. The quantum enhanced sensitivity of the states is combined with the subsequent application of a compound pulse sequence that allows us to separate atoms by several lattice sites. This, together with the capability to load small atomic clouds in the lattice at micrometric distances from a surface, make our setup ideal for sensing short-range forces. We show that for arrays of 10^{4} atoms, our protocol can reduce the required averaging time by a factor of 10 compared to unentangled lattice-based interferometers after accounting for primary sources of decoherence.

Scaling Laws for the Sensitivity Enhancement of Non-Gaussian Spin States

We identify the large-N scaling of the metrological quantum gain offered by over-squeezed spin states that are accessible by one-axis twisting, as a function of the preparation time. We further determine how the scaling is modified by relevant decoherence processes and predict a discontinuous change of the quantum gain at a critical preparation time that depends on the noise. Our analytical results provide recipes for optimal and feasible implementations of quantum enhancements with non-Gaussian spin states in existing experiments, well beyond the reach of spin squeezing.

Subradiant-to-Subradiant Phase Transition in the Bad Cavity Laser

We show that the onset of steady-state superradiance in a bad cavity laser is preceded by a dissipative phase transition between two distinct phases of steady-state subradiance. The transition is marked by a nonanalytic behavior of the cavity output power and the mean atomic inversion, as well as a discontinuity in the variance of the collective atomic inversion. In particular, for repump rates below a critical value, the cavity output power is strongly suppressed and does not increase with the atom number, while it scales linearly with atom number above this value. Remarkably, we find that the atoms are in a macroscopically entangled steady state near the critical region with a vanishing fraction of unentangled atoms in the large atom number limit.

Beating the standard quantum limit under ambient conditions with solid-state spins

The use of entangled sensors improves the precision limit from the standard quantum limit (SQL) to the Heisenberg limit. Most previous experiments beating the SQL are performed on the sensors that are well isolated under extreme conditions. Here, we demonstrate a sub-SQL interferometer at ambient conditions by using a multispin system, namely, the nitrogen-vacancy (NV) defect in diamond. We achieve two-spin interference with a phase sensitivity of 1.79 ± 0.06 dB beyond the SQL and three-spin interference with a phase sensitivity of 2.77 ± 0.10 dB. Besides, a magnetic sensitivity of 0.87 ± 0.09 dB beyond the SQL is achieved by two-spin interference for detecting a real magnetic field. Particularly, the deterministic and joint initialization of NV negative state, NV electron spin, and two nuclear spins is realized at room temperature. The techniques used here are of fundamental importance for quantum sensing and computing, and naturally applicable to other solid-state spin systems.

Metrological Detection of Multipartite Entanglement from Young Diagrams

We characterize metrologically useful multipartite entanglement by representing partitions with Young diagrams. We derive entanglement witnesses that are sensitive to the shape of Young diagrams and show that Dyson's rank acts as a resource for quantum metrology. Common quantifiers, such as the entanglement depth and k-separability are contained in this approach as the diagram's width and height. Our methods are experimentally accessible in a wide range of atomic systems, as we illustrate by analyzing published data on the quantum Fisher information and spin-squeezing coefficients.

Entanglement on an optical atomic-clock transition

State-of-the-art atomic clocks are based on the precise detection of the energy difference between two atomic levels, which is measured in terms of the quantum phase accumulated over a given time interval^1-4. The stability of optical-lattice clocks (OLCs) is limited both by the interrupted interrogation of the atomic system by the local-oscillator laser (Dick noise⁵) and by the standard quantum limit (SQL) that arises from the quantum noise associated with discrete measurement outcomes. Although schemes for removing the Dick noise have been recently proposed and implemented^4,6-8, performance beyond the SQL by engineering quantum correlations (entanglement) between atoms^9-20 has been demonstrated only in proof-of-principle experiments with microwave clocks of limited stability. The generation of entanglement on an optical-clock transition and operation of an OLC beyond the SQL represent important goals in quantum metrology, but have not yet been demonstrated experimentally¹⁶. Here we report the creation of a many-atom entangled state on an OLC transition, and use it to demonstrate a Ramsey sequence with an Allan deviation below the SQL after subtraction of the local-oscillator noise. We achieve a metrological gain of [Formula: see text] decibels over the SQL by using an ensemble consisting of a few hundred ytterbium-171 atoms, corresponding to a reduction of the averaging time by a factor of 2.8 ± 0.3. Our results are currently limited by the phase noise of the local oscillator and Dick noise, but demonstrate the possible performance improvement in state-of-the-art OLCs^1-4 through the use of entanglement. This will enable further advances in timekeeping precision and accuracy, with many scientific and technological applications, including precision tests of the fundamental laws of physics^21-23, geodesy^24-26 and gravitational-wave detection²⁷.

Prospects and challenges for squeezing-enhanced optical atomic clocks

Optical atomic clocks are a driving force for precision measurements due to the high accuracy and stability demonstrated in recent years. While further improvements to the stability have been envisioned by using entangled atoms, squeezing the quantum mechanical projection noise, evaluating the overall gain must incorporate essential features of an atomic clock. Here, we investigate the benefits of spin squeezed states for clocks operated with typical Brownian frequency noise-limited laser sources. Based on an analytic model of the closed servo-loop of an optical atomic clock, we report here quantitative predictions on the optimal clock stability for a given dead time and laser noise. Our analytic predictions are in good agreement with numerical simulations of the closed servo-loop. We find that for usual cyclic Ramsey interrogation of single atomic ensembles with dead time, even with the current most stable lasers spin squeezing can only improve the clock stability for ensembles below a critical atom number of about one thousand in an optical Sr lattice clock. Even with a future improvement of the laser performance by one order of magnitude the critical atom number still remains below 100,000. In contrast, clocks based on smaller, non-scalable ensembles, such as ion clocks, can already benefit from squeezed states with current clock lasers.

Retrodiction beyond the Heisenberg uncertainty relation

In quantum mechanics, the Heisenberg uncertainty relation presents an ultimate limit to the precision by which one can predict the outcome of position and momentum measurements on a particle. Heisenberg explicitly stated this relation for the prediction of “hypothetical future measurements”, and it does not describe the situation where knowledge is available about the system both earlier and later than the time of the measurement. Here, we study what happens under such circumstances with an atomic ensemble containing 10¹¹ rubidium atoms, initiated nearly in the ground state in the presence of a magnetic field. The collective spin observables of the atoms are then well described by canonical position and momentum observables, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\text{A}}$$\end{document}x^A and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{p}}_{\text{A}}$$\end{document}p^A that satisfy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[{\hat{x}}_{\text{A}},{\hat{p}}_{\text{A}}]=i\hslash$$\end{document}[x^A,p^A]=iℏ. Quantum non-demolition measurements of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{p}}_{\text{A}}$$\end{document}p^A before and of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\text{A}}$$\end{document}x^A after time t allow precise estimates of both observables at time t. By means of the past quantum state formalism, we demonstrate that outcomes of measurements of both the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{x}}_{\text{A}}$$\end{document}x^A and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{p}}_{A}$$\end{document}p^A observables can be inferred with errors below the standard quantum limit. The capability of assigning precise values to multiple observables and to observe their variation during physical processes may have implications in quantum state estimation and sensing.

If we have access to information about a quantum system both before and after a measurement, we are not in the usual remit of the Heisenberg uncertainty principle anymore. Here, the authors demonstrate that, in such a scenario, one can retrodict position and momentum measurements without being limited by HUR.

Enhancing Spin-Phonon and Spin-Spin Interactions Using Linear Resources in a Hybrid Quantum System

Hybrid spin-mechanical setups offer a versatile platform for quantum science and technology, but improving the spin-phonon as well as the spin-spin couplings of such systems remains a crucial challenge. Here, we propose and analyze an experimentally feasible and simple method for exponentially enhancing the spin-phonon and the phonon-mediated spin-spin interactions in a hybrid spin-mechanical setup, using only linear resources. Through modulating the spring constant of the mechanical cantilever with a time-dependent pump, we can acquire a tunable and nonlinear (two-phonon) drive to the mechanical mode, thus amplifying the mechanical zero-point fluctuations and directly enhancing the spin-phonon coupling. This method allows the spin-mechanical system to be driven from the weak-coupling regime to the strong-coupling regime, and even the ultrastrong coupling regime. In the dispersive regime, this method gives rise to a large enhancement of the phonon-mediated spin-spin interactions between distant solid-state spins, typically two orders of magnitude larger than that without modulation. As an example, we show that the proposed scheme can apply to generating entangled states of multiple spins with high fidelities even in the presence of large dissipations.

Free Space Ramsey Spectroscopy in Rubidium with Noise below the Quantum Projection Limit

We demonstrate the utility of optical cavity generated spin-squeezed states in free space atomic fountain clocks in ensembles of 390 000 ^{87}Rb atoms. Fluorescence imaging, correlated to an initial quantum nondemolition measurement, is used for population spectroscopy after the atoms are released from a confining lattice. For a free fall time of 4 milliseconds, we resolve a single-shot phase sensitivity of 814(61) microradians, which is 5.8(0.6) decibels (dB) below the quantum projection limit. We observe that this squeezing is preserved as the cloud expands to a roughly 200  μm radius and falls roughly 300  μm in free space. Ramsey spectroscopy with 240 000 atoms at a 3.6 ms Ramsey time results in a single-shot fractional frequency stability of 8.4(0.2)×10^{-12}, 3.8(0.2) dB below the quantum projection limit. The sensitivity and stability are limited by the technical noise in the fluorescence detection protocol and the microwave system, respectively.

Mechanical Squeezing via Fast Continuous Measurement

We revisit quantum state preparation of an oscillator by continuous linear position measurement. Quite general analytical expressions are derived for the conditioned state of the oscillator. Remarkably, we predict that quantum squeezing is possible outside of both the backaction dominated and quantum coherent oscillation regimes, relaxing experimental requirements even compared to ground-state cooling. This provides a new way to generate nonclassical states of macroscopic mechanical oscillators, and opens the door to quantum sensing and tests of quantum macroscopicity at room temperature.

Generating Greenberger-Horne-Zeilinger states with squeezing and postselection

Many quantum state preparation methods rely on a combination of dissipative quantum state initialization followed by unitary evolution to a desired target state. Here we demonstrate the usefulness of quantum measurement as an additional tool for quantum state preparation. Starting from a pure separable multipartite state, a control sequence, which includes rotation, spin squeezing via one-axis twisting, quantum measurement, and postselection, generates highly entangled multipartite states, which we refer to as projected squeezed (PS) states. Through an optimization method, we then identify parameters required to maximize the overlap fidelity of the PS states with the maximally entangled Greenberger-Horne-Zeilinger (GHZ) states. The method leads to an appreciable decrease in the state preparation time of GHZ states for successfully postselected outcomes when compared to preparation through unitary evolution with one-axis twisting only.

Protocol for Precise Field Sensing in the Optical Domain with Cold Atoms in a Cavity

In the context of quantum metrology, optical cavity-QED platforms have primarily been focused on the generation of entangled atomic spin states useful for next-generation frequency and time standards. Here, we report a complementary application: the use of optical cavities to generate nonclassical states of light for electric field sensing below the standard quantum limit. We show that cooperative atom-light interactions in the strong collective coupling regime can be used to engineer generalized atom-light cat states which enable quantum enhanced sensing of small displacements of the cavity field even in the presence of photon loss. We demonstrate that metrological gains of 10-20 dB below the standard quantum limit are within reach for current cavity-QED systems operating with long-lived alkaline-earth atoms.

Spin squeezing of 1011 atoms by prediction and retrodiction measurements

The measurement sensitivity of quantum probes using N uncorrelated particles is restricted by the standard quantum limit¹, which is proportional to [Formula: see text]. This limit, however, can be overcome by exploiting quantum entangled states, such as spin-squeezed states². Here we report the measurement-based generation of a quantum state that exceeds the standard quantum limit for probing the collective spin of 10¹¹ rubidium atoms contained in a macroscopic vapour cell. The state is prepared and verified by sequences of stroboscopic quantum non-demolition (QND) measurements. We then apply the theory of past quantum states^3,4 to obtain spin state information from the outcomes of both earlier and later QND measurements. Rather than establishing a physically squeezed state in the laboratory, the past quantum state represents the combined system information from these prediction and retrodiction measurements. This information is equivalent to a noise reduction of 5.6 decibels and a metrologically relevant squeezing of 4.5 decibels relative to the coherent spin state. The past quantum state yields tighter constraints on the spin component than those obtained by conventional QND measurements. Our measurement uses 1,000 times more atoms than previous squeezing experiments^5-10, with a corresponding angular variance of the squeezed collective spin of 4.6 × 10^-13 radians squared. Although this work is rooted in the foundational theory of quantum measurements, it may find practical use in quantum metrology and quantum parameter estimation, as we demonstrate by applying our protocol to quantum enhanced atomic magnetometry.

Cavity-enhanced non-destructive detection of atoms for an optical lattice clock

We demonstrate a new method of cavity-enhanced non-destructive detection of atoms for a strontium optical lattice clock. The detection scheme is shown to be linear in atom number up to at least 2×10⁴ atoms, to reject technical noise sources, to achieve signal to noise ratio close to the photon shot noise limit, to provide spatially uniform atom-cavity coupling, and to minimize inhomogeneous ac Stark shifts. These features enable detection of atoms with minimal perturbation to the atomic state, a critical step towards realizing an ultra-high-stability, quantum-enhanced optical lattice clock.

Continuous Real-Time Tracking of a Quantum Phase Below the Standard Quantum Limit

We propose a scheme for continuously measuring the evolving quantum phase of a collective spin composed of N pseudospins. Quantum nondemolition measurements of a lossy cavity mode interacting with an atomic ensemble are used to directly probe the phase of the collective atomic spin without converting it into a population difference. Unlike traditional Ramsey measurement sequences, our scheme allows for real-time tracking of time-varying signals. As a bonus, spin-squeezed states develop naturally, providing real-time phase estimation significantly more precise than the standard quantum limit of Δϕ_{SQL}=1/sqrt[N]  rad.

Near-Unitary Spin Squeezing in ^{171}Yb

Spin squeezing can improve atomic precision measurements beyond the standard quantum limit (SQL), and unitary spin squeezing is essential for improving atomic clocks. We report substantial and nearly unitary spin squeezing in ^{171}Yb, an optical lattice clock atom. The collective nuclear spin of ∼10^{3} atoms is squeezed by cavity feedback, using light detuned from the system's resonances to attain unitarity. The observed precision gain over the SQL is limited by state readout to 6.5(4) dB, while the generated states offer a gain of 12.9(6) dB, limited by the curvature of the Bloch sphere. Using a squeezed state within 30% of unitarity, we demonstrate an interferometer that improves the averaging time over the SQL by a factor of 3.7(2). In the future, the squeezing can be simply transferred onto the optical-clock transition of ^{171}Yb.

Metrological Nonlinear Squeezing Parameter

The well-known metrological linear squeezing parameters (such as quadrature or spin squeezing) efficiently quantify the sensitivity of Gaussian states. Yet, these parameters are insufficient to characterize the much wider class of highly sensitive non-Gaussian states. Here, we introduce a class of metrological nonlinear squeezing parameters obtained by analytical optimization of measurement observables among a given set of accessible (possibly nonlinear) operators. This allows for the metrological characterization of non-Gaussian quantum states of discrete and continuous variables. Our results lead to optimized and experimentally feasible recipes for a high-precision moment-based estimation of a phase parameter and can be used to systematically construct multipartite entanglement and nonclassicality witnesses for complex quantum states.

Experimentally reducing the quantum measurement back action in work distributions by a collective measurement

In quantum thermodynamics, the standard approach to estimating work fluctuations in unitary processes is based on two projective measurements, one performed at the beginning of the process and one at the end. The first measurement destroys any initial coherence in the energy basis, thus preventing later interference effects. To decrease this back action, a scheme based on collective measurements has been proposed by Perarnau-Llobet et al. Here, we report its experimental implementation in an optical system. The experiment consists of a deterministic collective measurement on two identically prepared qubit states, encoded in the polarization and path degree of a single photon. The standard two-projective measurement approach is also experimentally realized for comparison. Our results show the potential of collective schemes to decrease the back action of projective measurements, and capture subtle effects arising from quantum coherence.

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.

Exploring gravity with the MIGA large scale atom interferometer

We present the MIGA experiment, an underground long baseline atom interferometer to study gravity at large scale. The hybrid atom-laser antenna will use several atom interferometers simultaneously interrogated by the resonant mode of an optical cavity. The instrument will be a demonstrator for gravitational wave detection in a frequency band (100 mHz–1 Hz) not explored by classical ground and space-based observatories, and interesting for potential astrophysical sources. In the initial instrument configuration, standard atom interferometry techniques will be adopted, which will bring to a peak strain sensitivity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{2}}\cdot {\bf{1}}{{\bf{0}}}^{-{\bf{13}}}/\sqrt{{\bf{H}}{\bf{z}}}$$\end{document}2⋅10−13/Hz at 2 Hz. This demonstrator will enable to study the techniques to push further the sensitivity for the future development of gravitational wave detectors based on large scale atom interferometers. The experiment will be realized at the underground facility of the Laboratoire Souterrain à Bas Bruit (LSBB) in Rustrel–France, an exceptional site located away from major anthropogenic disturbances and showing very low background noise. In the following, we present the measurement principle of an in-cavity atom interferometer, derive the method for Gravitational Wave signal extraction from the antenna and determine the expected strain sensitivity. We then detail the functioning of the different systems of the antenna and describe the properties of the installation site.

Robust Spin Squeezing via Photon-Mediated Interactions on an Optical Clock Transition

Cavity QED is a promising avenue for the deterministic generation of entangled and spin-squeezed states for quantum metrology. One archetypal scheme generates squeezing via collective one-axis twisting interactions. However, we show that in implementations using optical transitions in long-lived atoms the achievable squeezing is fundamentally limited by collectively enhanced emission into the cavity mode which is generated in parallel with the cavity-mediated spin-spin interactions. We propose an alternative scheme which generates a squeezed state that is protected from collective emission, and investigate its sensitivity to realistic sources of experimental noise and imperfections.

Efficient Adiabatic Spin-Dependent Kicks in an Atom Interferometer

We present an atom interferometry technique in which the beam splitter is split into two separate operations. A microwave pulse first creates a spin-state superposition, before optical adiabatic passage spatially separates the arms of that superposition. Despite using a thermal atom sample in a small (600  μm) interferometry beam, this procedure delivers an efficiency of 99% per ℏk of momentum separation. Utilizing this efficiency, we first demonstrate interferometry with up to 16ℏk momentum splitting and free-fall limited interrogation times. We then realize a single-source gradiometer, in which two interferometers measuring a relative phase originate from the same atomic wave function. Finally, we demonstrate a resonant interferometer with over 100 adiabatic passages, and thus over 400ℏk total momentum transferred.

Beating the classical precision limit with spin-1 Dicke states of more than 10,000 atoms

Interferometry is a paradigm for most precision measurements. Using N uncorrelated particles, the achievable precision for a two-mode (two-path) interferometer is bounded by the standard quantum limit (SQL), [Formula: see text], due to the discrete (quanta) nature of individual measurements. Despite being a challenging benchmark, the two-mode SQL has been approached in a number of systems, including the Laser Interferometer Gravitational-Wave Observatory and today's best atomic clocks. For multimode interferometry, the SQL becomes [Formula: see text] using M modes. Higher precision can also be achieved using entangled particles such that quantum noises from individual particles cancel out. In this work, we demonstrate an interferometric precision of [Formula: see text] dB beyond the three-mode SQL, using balanced spin-1 (three-mode) Dicke states containing thousands of entangled atoms. The input quantum states are deterministically generated by controlled quantum phase transition and exhibit close to ideal quality. Our work shines light on the pursuit of quantum metrology beyond SQL.

Arbitrary digital pulse sequence generator with delay-loop timing

We propose an idea of an electronic multi-channel arbitrary digital sequence generator with temporal granularity equal to two clock cycles. We implement the generator with 32 channels using a low-cost ARM microcontroller and demonstrate its capability to produce temporal delays ranging from tens of nanoseconds to hundreds of seconds, with 24 ns timing granularity and linear scaling of delay with respect to the number of delay loop iterations. The generator is optionally synchronized with an external clock source to provide 100 ps jitter and overall sequence repeatability within the whole temporal range. The generator is fully programmable and able to produce digital sequences of high complexity. The concept of the generator can be implemented using different microcontrollers and applied for controlling of various optical, atomic, and nuclear physics measurement setups.

Squeezing on Momentum States for Atom Interferometry

We propose and analyze a method that allows for the production of squeezed states of the atomic center-of-mass motion that can be injected into an atom interferometer. Our scheme employs dispersive probing in a ring resonator on a narrow transition in order to provide a collective measurement of the relative population of two momentum states. We show that this method is applicable to a Bragg diffraction-based strontium atom interferometer with large diffraction orders. This technique can be extended also to small diffraction orders and large atom numbers N by inducing atomic transparency at the frequency of the probe field, reaching an interferometer phase resolution scaling Δϕ∼N^{-3/4}. We show that for realistic parameters it is possible to obtain a 20 dB gain in interferometer phase estimation compared to the standard quantum limit. Our method is applicable to other atomic species where a narrow transition is available or can be synthesized.

An atom interferometer inside a hollow-core photonic crystal fiber

Coherent interactions between electromagnetic and matter waves lie at the heart of quantum science and technology. However, the diffraction nature of light has limited the scalability of many atom-light-based quantum systems. We use the optical fields in a hollow-core photonic crystal fiber to spatially split, reflect, and recombine a coherent superposition state of free-falling ⁸⁵Rb atoms to realize an inertia-sensitive atom interferometer. The interferometer operates over a diffraction-free distance, and the contrasts and phase shifts at different distances agree within one standard error. The integration of phase coherent photonic and quantum systems here shows great promise to advance the capability of atom interferometers in the field of precision measurement and quantum sensing with miniature design of apparatus and high efficiency of laser power consumption.

Cavity-Assisted Measurement and Coherent Control of Collective Atomic Spin Oscillators

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 backaction heating, it achieves a steady-state temperature, indicated by an asymmetry between the Stokes and the 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.

Preparation of Ultracold Atom Clouds at the Shot Noise Level

We prepare number stabilized ultracold atom clouds through the real-time analysis of nondestructive images and the application of feedback. In our experiments, the atom number N∼10^{6} is determined by high precision Faraday imaging with uncertainty ΔN below the shot noise level, i.e., ΔN<sqrt[N]. Based on this measurement, feedback is applied to reduce the atom number to a user-defined target, whereupon a second imaging series probes the number stabilized cloud. By this method, we show that the atom number in ultracold clouds can be prepared below the shot noise level.

Quantum spin dynamics and entanglement generation with hundreds of trapped ions

Quantum simulation of spin models can provide insight into problems that are difficult or impossible to study with classical computers. Trapped ions are an established platform for quantum simulation, but only systems with fewer than 20 ions have demonstrated quantum correlations. We studied quantum spin dynamics arising from an engineered, homogeneous Ising interaction in a two-dimensional array of (9)Be(+) ions in a Penning trap. We verified entanglement in spin-squeezed states of up to 219 ions, directly observing 4.0 ± 0.9 decibels of spectroscopic enhancement, and observed states with non-Gaussian statistics consistent with oversqueezed states. The good agreement with ab initio theory that includes interactions and decoherence lays the groundwork for simulations of the transverse-field Ising model with variable-range interactions, which are generally intractable with classical methods.