We demonstrate the creation of entangled, spin-squeezed states using a collective, or joint, measurement and real-time feedback. The pseudospin state of an ensemble of N = 5 × 10 4 laser-cooled 87 Rb atoms is deterministically driven to a specified population state with angular resolution that is a factor of 5.5(8) [7.4(6) dB] in variance below the standard quantum limit for unentangled atomscomparable to the best enhancements using only unitary evolution. Without feedback, conditioning on the outcome of the joint premeasurement, we directly observe up to 59(8) times [17.7( 6) dB] improvement in quantum phase variance relative to the standard quantum limit for N = 4 × 10 5 atoms. This is one of the largest reported entanglement enhancements to date in any system.

We demonstrate a method to generate spatially homogeneous entangled, spin-squeezed states of atoms appropriate for maintaining a large amount of squeezing even after release into the arm of a matter-wave interferometer or other free space quantum sensor. Using an effective intracavity dipole trap, we allow atoms to move along the cavity axis and time average their coupling to the standing wave used to generate entanglement via collective measurements, demonstrating 11(1) dB of directly observed spin squeezing. Our results show that time averaging in collective measurements can greatly reduce the impact of spatially inhomogeneous coupling to the measurement apparatus.Spin-1/2 atoms must project into either "up" or "down" when measured. For N unentangled atoms, the independent randomness in this quantum projection fundamentally limits the single-shot phase resolution of any quantum sensor to ∆φ SQL = 1/ √ N rad, the standard quantum limit (SQL) [1]. Collective measurements of atoms in optical cavities have recently produced some of the most strongly entangled, spin-squeezed states to date, directly improving the phase resolution of a quantum sensor's "clock hand" by a factor up to 60-70 (roughly 18 dB) in noise variance below the SQL [2,3].Spin-squeezed states could be used to improve a wide range of quantum sensors, with today's best atomic clocks [4][5][6] being particularly promising candidates [7,8]. In this work we focus on preparing spin-squeezed states appropriate for matter-wave atom interferometry with applications including inertial sensing [9], measurements of gravity and freefall, [10,11] and even the search for certain proposed types of dark matter and dark energy [12,13].A major challenge arises for cavity-based atom interferometry and other applications involving release of spinsqueezed atoms into free space. The problem is that the probe mode used to perform the collective measurement is a standing wave, but the atoms are trapped in a 1-dimensional lattice defined by a standing wave cavity mode with a significantly different wavelength. Some atoms will sit in lattice sites positioned near nodes and some near anti-nodes of the entanglement-generating probe light. As a result, the atoms will contribute to the collective measurement with different strengths. In this common case, the large degree of squeezing exists only for this specific coupling configuration and would be largely lost after releasing the atoms into the arm of an interferometer, since their final coupling to the cavity mode or other readout detector will be different from the original configuration [14]. In contrast, we wish to create spatially homogeneous entanglement, quantified by the amount of observed phase resolution beyond the SQL that one can achieve when every atom couples equally to the final measurement apparatus.In this Letter, we demonstrate a method to create ho- FIG. 1. (a)Optical lattice sidebands separated by one free spectral range (FSR) are injected into the cavity to create an axially homogeneous "dipole" trap. Dip...

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 limit1–3 set by projections of individual atoms. Large amounts of entanglement4–6 involving the internal degrees of freedom of laser-cooled atomic ensembles4–16 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 $$3\,.\,{4}_{-0.9}^{+1.1}$$ 3 . 4 − 0.9 + 1.1  dB and $$2\,.\,{5}_{-0.6}^{+0.6}$$ 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 $$1\,.\,{7}_{-0.5}^{+0.5}$$ 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 sensors17,18, searches for new physics, particles and fields19–23, future advanced gravitational wave detectors24,25 and accessing beyond mean-field quantum many-body physics26–30.

We propose a scheme for continuously measuring the evolving quantum phase of a collective spin composed of N pseudospins. Quantum non-demolition 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, spinsqueezed states develop naturally, providing real-time phase estimation significantly more precise than the standard quantum limit of ∆φSQL = 1/ √ N radians.Quantum systems have become robust platforms for metrology and tests of fundamental physics. Many applications rely on the dynamics of pseudospin-1/2 systems with two long-lived quantum states, |↑ and |↓ . After preparing an equal superposition of these two states, a physical interaction is studied by investigating its effect on the relative phase φ(t), with the state of each spin evolving in time as |ψ(t) = |↓ + e iφ(t) |↑ / √ 2. We propose a novel scheme that enables continuous tracking of this relative phase. Our scheme continuously and directly measures the real-time phase φ(t) unlike the widely used Ramsey sequence [1][2][3][4][5][6][7][8][9][10][11][12], which indirectly measures the net accumulated phase φ(T ) during an interrogation time T . The typically destructive readout in a Ramsey sequence requires multiple state resets, rotations and repetitions of the sequence to infer the phase at different times from a population difference. In contrast, a single run of our protocol yields a continuous time series of phase measurements. Therefore, our scheme enables real-time tracking of time-varying signals that are not reproducible.As an added benefit, our scheme yields continuous phase estimates with precision well beyond the standard quantum limit (SQL) of ∆φ SQL = 1/ √ N radians that limits readout precision with N unentangled spins. In comparison to several proposals and experiments [13][14][15][16][17][18][19] that have demonstrated squeezed states with precision beyond the SQL, our scheme enjoys the advantage that the squeezing is produced, the phase accumulated, and the readout performed, all in the same spin quadrature.Recent experiments have demonstrated phase tracking of a spin using quantum non-demolition (QND) measurements via a Faraday rotation angle [20]. In contrast, our proposal is based on interfering Raman transitions in a cavity and enables an intuitive interpretation of phase tracking in terms of elementary atom-cavity interactions that nearly balance one another. Our scheme directly reveals a phasor precessing in the equatorial plane of a Bloch sphere, in the spirit of the "hand on a clock" analogy at the core of quantum metrology.We represent the collective angular momentum of N atomic spins by a classical Bloch vector of length N/2 FIG. 1. Schematic and working principle. (a) Two lasers drive a collection of atoms to interact with a cavity mode. The relative phase φ(t) can be...