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 interval1-4. 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 implemented4,6-8, performance beyond the SQL by engineering quantum correlations (entanglement) between atoms9-20 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 OLCs1-4 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 physics21-23, geodesy24-26 and gravitational-wave detection27.

Using an acousto-optic modulator as a fast spatial light modulator.

High-speed spatial light modulators (SLM) are crucial components for free-space communication and structured illumination imaging. Current approaches for dynamical spatial mode generation, such as liquid crystal SLMs or digital micromirror devices, are limited to a maximum pattern refresh rate of 10 kHz and have a low damage threshold. We demonstrate that arbitrary spatial profiles in a laser pulse can be generated by mapping the temporal radio-frequency (RF) waveform sent to an acousto-optic modulator (AOM) onto the optical field. We find that the fidelity of the SLM performance can be improved through numerical optimization of the RF waveform to overcome the nonlinear effect of AOM. An AOM can thus be used as a 1-dimensional SLM, a technique we call acousto-optic spatial light modulator (AO-SLM), which has 50 µm pixel pitch, over 1 MHz update rate, and high damage threshold. We simulate the application of AO-SLM to single-pixel imaging, which can reconstruct a 32×32 pixel complex object at a rate of 11.6 kHz with 98% fidelity.

High finesse bow-tie cavity for strong atom-photon coupling in Rydberg arrays.

We report a high-finesse bow-tie cavity designed for atomic physics experiments with Rydberg atom arrays. The cavity has a finesse of 51,000 and a waist of 7.1 µm at the cesium D2 line (852 nm). With these parameters, the cavity is expected to induce strong coupling between a single atom and a single photon, corresponding to a cooperativity per traveling mode of 35 at the cavity waist. To trap and image atoms, the cavity setup utilizes two in-vacuum aspheric lenses with a numerical aperture (NA) of 0.35 and is capable of housing NA = 0.5 microscope objectives. In addition, the large atom-mirror distance (≳1.5 cm) provides good optical access and minimizes stray electric fields at the position of the atoms. This cavity setup can operate in tandem with a Rydberg array platform, creating a fully connected system for quantum simulation and computation.

Confocal super-resolution microscopy based on a spatial mode sorter.

Spatial resolution is one of the most important specifications of an imaging system. Recent results in the quantum parameter estimation theory reveal that an arbitrarily small distance between two incoherent point sources can always be efficiently determined through the use of a spatial mode sorter. However, extending this procedure to a general object consisting of many incoherent point sources remains challenging, due to the intrinsic complexity of multi-parameter estimation problems. Here, we generalize the Richardson-Lucy (RL) deconvolution algorithm to address this challenge. We simulate its application to an incoherent confocal microscope, with a Zernike spatial mode sorter replacing the pinhole used in a conventional confocal microscope. We test different spatially incoherent objects of arbitrary geometry, and we find that the resolution enhancement of sorter-based microscopy is on average over 30% higher than that of a conventional confocal microscope using the standard RL deconvolution algorithm. Our method could potentially be used in diverse applications such as fluorescence microscopy and astronomical imaging.

Fast generation and detection of spatial modes of light using an acousto-optic modulator.

Spatial modes of light provide a high-dimensional space that can be used to encode both classical and quantum information. Current approaches for dynamically generating and measuring these modes are slow, due to the need to reconfigure a high-resolution phase mask such as a spatial light modulator or digital micromirror device. The process of updating the spatial mode of light can be greatly accelerated by multiplexing a set of static phase masks with a fast, image-preserving optical switch, such as an acousto-optic modulator (AOM). We experimentally realize this approach, using a double-pass AOM to generate one of five orbital angular momentum states with a switching rate of up to 500 kHz. We then apply this system to perform fast quantum state tomography of spatial modes of light in a 2-dimensional Hilbert space by projecting the unknown state onto six spatial modes comprising three mutually unbiased bases. We are able to reconstruct arbitrary states in under 1 ms with an average fidelity of 96.9%.

Performance of real-time adaptive optics compensation in a turbulent channel with high-dimensional spatial-mode encoding.

The orbital angular momentum (OAM) of photons is a promising degree of freedom for high-dimensional quantum key distribution (QKD). However, effectively mitigating the adverse effects of atmospheric turbulence is a persistent challenge in OAM QKD systems operating over free-space communication channels. In contrast to previous works focusing on correcting static simulated turbulence, we investigate the performance of OAM QKD in real atmospheric turbulence with real-time adaptive optics (AO) correction. We show that even though our AO system provides a limited correction, it is possible to mitigate the errors induced by weak turbulence and establish a secure channel. The crosstalk induced by turbulence and the performance of AO systems is investigated in two configurations: a lab-scale link with controllable turbulence, and a 340 m long cross-campus link with dynamic atmospheric turbulence. Our experimental results suggest that an advanced AO system with fine beam tracking, reliable beam stabilization, precise wavefront sensing, and accurate wavefront correction is necessary to adequately correct turbulence-induced error. We also propose and demonstrate different solutions to improve the performance of OAM QKD with turbulence, which could enable the possibility of OAM encoding in strong turbulence.

Robust kHz-linewidth distributed Bragg reflector laser with optoelectronic feedback.

We demonstrate a combination of optical and electronic feedback that significantly narrows the linewidth of distributed Bragg reflector lasers (DBRs). We use optical feedback from a long external fiber path to reduce the high-frequency noise of the laser. An electro-optic modulator placed inside the optical feedback path allows us to apply electronic feedback to the laser frequency with very large bandwidth, enabling robust and stable locking to a reference cavity that suppresses low-frequency components of laser noise. The combination of optical and electronic feedback allows us to significantly lower the frequency noise power spectral density of the laser across all frequencies and narrow its linewidth from a free-running value of 1.1 MHz to a stabilized value of 1.9 kHz, limited by the detection system resolution. This approach enables the construction of robust lasers with sub-kHz linewidth based on DBRs across a broad range of wavelengths.

Quantum Nonlocal Aberration Cancellation.

Phase distortions, or aberrations, can negatively influence the performance of an optical imaging system. Through the use of position-momentum entangled photons, we nonlocally correct for aberrations in one photon's optical path by intentionally introducing the complementary aberrations in the optical path of the other photon. In particular, we demonstrate the simultaneous nonlocal cancellation of aberrations that are of both even and odd order in the photons' transverse degrees of freedom. We also demonstrate a potential application of this technique by nonlocally canceling the effect of defocus in a quantum imaging experiment and thereby recover the original spatial resolution.

Near-Unitary Spin Squeezing in

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.

Preventing and reversing vacuum-induced optical losses in high-finesse tantalum (V) oxide mirror coatings.

High-finesse optical cavities placed under vacuum are foundational platforms in quantum information science with photons and atoms. We study the vacuum-induced degradation of high-finesse optical cavities with mirror coatings composed of SiO2-Ta2O5 dielectric stacks, and present methods to protect these coatings and to recover their initial low loss levels. For separate coatings with reflectivities centered at 370 nm and 422 nm, a vacuum-induced continuous increase in optical loss occurs if the surface-layer coating is made of Ta2O5, while it does not occur if it is made of SiO2. The incurred optical loss can be reversed by filling the vacuum chamber with oxygen at atmospheric pressure, and the recovery rate can be strongly accelerated by continuous laser illumination at 422 nm. Both the degradation and the recovery processes depend strongly on temperature. We find that a 1 nm-thick layer of SiO2 passivating the Ta2O5 surface layer is sufficient to reduce the degradation rate by more than a factor of 10, strongly supporting surface oxygen depletion as the primary degradation mechanism.

Carving Complex Many-Atom Entangled States by Single-Photon Detection.

We propose a versatile and efficient method to generate a broad class of complex entangled states of many atoms via the detection of a single photon. For an atomic ensemble contained in a strongly coupled optical cavity illuminated by weak single- or multifrequency light, the atom-light interaction entangles the frequency spectrum of a transmitted photon with the collective spin of the atomic ensemble. Simple time-resolved detection of the transmitted photon then projects the atomic ensemble into a desired pure entangled state. This method can be implemented with existing technology, yields high success probability per trial, and can generate complex entangled states such as mesoscopic superposition states of coherent spin states with high fidelity.

Passive intrinsic-linewidth narrowing of ultraviolet extended-cavity diode laser by weak optical feedback.

We present a simple method for narrowing the intrinsic Lorentzian linewidth of a commercial ultraviolet grating extended-cavity diode laser (TOPTICA DL Pro) using weak optical feedback from a long external cavity. We achieve a suppression in frequency noise spectral density of 20 dB measured at frequencies around 1 MHz, corresponding to the narrowing of the intrinsic Lorentzian linewidth from 200 kHz to 2 kHz. Provided additional active low-frequency noise suppression and long-term drift compensation, the system is suitable for experiments requiring a tunable ultraviolet laser with narrow linewidth and low high-frequency noise, such as precision spectroscopy, optical clocks, and quantum information science experiments.

Proposal to observe the nonlocality of Bohmian trajectories with entangled photons.

Bohmian mechanics reproduces all statistical predictions of quantum mechanics, which ensures that entanglement cannot be used for superluminal signaling. However, individual Bohmian particles can experience superluminal influences. We propose to illustrate this point using a double double-slit setup with path-entangled photons. The Bohmian velocity field for one of the photons can be measured using a recently demonstrated weak-measurement technique. The found velocities strongly depend on the value of a phase shift that is applied to the other photon, potentially at spacelike separation.

High-fidelity spatial mode transmission through a 1-km-long multimode fiber via vectorial time reversal.

The large number of spatial modes supported by standard multimode fibers is a promising platform for boosting the channel capacity of quantum and classical communications by orders of magnitude. However, the practical use of long multimode fibers is severely hampered by modal crosstalk and polarization mixing. To overcome these challenges, we develop and experimentally demonstrate a vectorial time reversal technique, which is accomplished by digitally pre-shaping the wavefront and polarization of the forward-propagating signal beam to be the phase conjugate of an auxiliary, backward-propagating probe beam. Here, we report an average modal fidelity above 80% for 210 Laguerre-Gauss and Hermite-Gauss modes by using vectorial time reversal over an unstabilized 1-km-long fiber. We also propose a practical and scalable spatial-mode-multiplexed quantum communication protocol over long multimode fibers to illustrate potential applications that can be enabled by our technique.

Note: Fast compact laser shutter using a direct current motor and three-dimensional printing.

We present a mechanical laser shutter design that utilizes a direct current electric motor to rotate a blade which blocks and unblocks a light beam. The blade and the main body of the shutter are modeled with computer aided design (CAD) and are produced by 3D printing. Rubber flaps are used to limit the blade's range of motion, reducing vibrations and preventing undesirable blade oscillations. At its nominal operating voltage, the shutter achieves a switching speed of (1.22 ± 0.02) m/s with 1 ms activation delay and 10 µs jitter in its timing performance. The shutter design is simple, easy to replicate, and highly reliable, showing no failure or degradation in performance over more than 10(8) cycles.

Scale-specific multifractal medical image analysis.

Fractal geometry has been applied widely in the analysis of medical images to characterize the irregular complex tissue structures that do not lend themselves to straightforward analysis with traditional Euclidean geometry. In this study, we treat the nonfractal behaviour of medical images over large-scale ranges by considering their box-counting fractal dimension as a scale-dependent parameter rather than a single number. We describe this approach in the context of the more generalized Rényi entropy, in which we can also compute the information and correlation dimensions of images. In addition, we describe and validate a computational improvement to box-counting fractal analysis. This improvement is based on integral images, which allows the speedup of any box-counting or similar fractal analysis algorithm, including estimation of scale-dependent dimensions. Finally, we applied our technique to images of invasive breast cancer tissue from 157 patients to show a relationship between the fractal analysis of these images over certain scale ranges and pathologic tumour grade (a standard prognosticator for breast cancer). Our approach is general and can be applied to any medical imaging application in which the complexity of pathological image structures may have clinical value.

A comparison of phase, amplitude, and velocity binning for cone-beam computed tomographic projection-based motion reconstruction.

PURPOSE: We previously developed a motion estimation technique based on direct cone-beam projection analysis. It is able to reconstruct the complete motion trajectory of a radio-opaque marker, including cycle-to-cycle variability, using respiratory binning of the projection images. This paper investigates the use of phase, amplitude, and amplitude-velocity binning in the context of projection-based cone-beam motion estimation (CBME). METHODS AND MATERIALS: We simulated cone-beam computed tomographic scans of 160 tumor trajectories estimated by a CyberKnife Synchrony System (Accuray, Sunnyvale, CA), and reconstructed the complete trajectory with CBME using phase, amplitude, and amplitude-velocity binning of the projection data. Various numbers of respiratory bins, from 1 (no binning) to 100, were used for phase and amplitude binning, while 1 to 100 amplitude bins with 4 velocity bins were used for amplitude-velocity binning. From this large pool of data, we correlated the reconstruction accuracy with bin type, total number of bins, number of breathing cycles per bin, and the position of the bin within the breathing cycle. RESULTS: CBME predicted the true motion of the marker with a 3-dimensional (3D) mean root mean square (RMS) error of 0.24 mm for amplitude-velocity binning, 0.31 mm for amplitude binning, and 0.52 mm for phase binning. Reconstruction 3D RMS error increased to over 1 mm when less than 3 breathing cycles contributed to a bin. We found that reconstruction accuracy was optimized when about 20 bins were used. Accuracy also decreased in bins located around the inhale portion of the breath cycle, compared with the mid- and end-exhale positions. CONCLUSIONS: This study provides a quantitative assessment of phase, amplitude, and amplitude-velocity binning for CBME. A joint binning approach should be used to give both the accuracy of amplitude binning, as well as the robustness of phase binning, in areas of limited motion sampling.

Observing the average trajectories of single photons in a two-slit interferometer.

A consequence of the quantum mechanical uncertainty principle is that one may not discuss the path or "trajectory" that a quantum particle takes, because any measurement of position irrevocably disturbs the momentum, and vice versa. Using weak measurements, however, it is possible to operationally define a set of trajectories for an ensemble of quantum particles. We sent single photons emitted by a quantum dot through a double-slit interferometer and reconstructed these trajectories by performing a weak measurement of the photon momentum, postselected according to the result of a strong measurement of photon position in a series of planes. The results provide an observationally grounded description of the propagation of subensembles of quantum particles in a two-slit interferometer.