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Entangled graph states can be used for quantum sensing and computing applications. Error correction in measurement-based quantum computing schemes will require the construction of cluster states in at least 3 dimensions. Here we generate 1-, 2-, 3-, and 4-dimensional optical frequency-mode cluster states by sending broadband 2-mode vacuum-squeezed light through an electro-optical modulator (EOM) driven with multiple frequencies. We create the squeezed light using 4-wave mixing in Rb atomic vapor and mix the sideband frequencies (qumodes) using an EOM, as proposed by Zhu et al. (1), producing a pattern of entanglement correlations that constitute continuous-variable graph states containing up to several hundred qumodes. We verify the entanglement structure by using homodyne measurements to construct the covariance matrices and evaluate the nullifiers. This technique enables scaling of optical cluster states to multiple dimensions without increasing loss.

High-precision measurements are crucial for testing the fundamental laws of nature and for advancing the technological frontier. Clock interferometry, where particles with an internal clock are coherently split and recombined along two spatial paths, has sparked significant interest due to its fundamental implications, especially at the intersection of quantum mechanics and general relativity. Here, we demonstrate that a clock interferometer provides metrological improvement with respect to its technical-noise-limited counterpart employing a single internal quantum state. This enhancement around a critical working point can be interpreted as a geometric-phase-induced signal-to-noise ratio gain. In our experimental setup, we infer a precision enhancement of 8.8 decibels when measuring a small difference between external fields. We estimate that tens of decibels of precision enhancement could be attained for measurements with a higher atom flux. This opens the door to the development of a superior probe for fundamental physics as well as a high-performance sensor for various technological applications.

We present a study of homodyne measurements of two-mode, vacuum-seeded, quadrature-squeezed light generated by four-wave mixing in warm rubidium vapor. Our results reveal that the vacuum squeezing can extend down to measurement frequencies of less than 1 Hz, and the squeezing bandwidth, similar to the seeded intensity-difference squeezing measured in this system, reaches up to approximately 20 MHz for typical pump parameters. By dividing the squeezing bandwidth into smaller frequency bins, we show that different sideband frequencies represent independent sources of two-mode squeezing. Such frequency bins may provide useful qumodes for quantum information processing experiments. We also investigate the impact of group velocity delays on the correlations in the system.

We investigate experimentally the nonlocal phase modulation of multiple-frequency-mode, continuous-variable entangled twin beams. We use a pair of electro-optical phase modulators to modulate the entangled probe and conjugate light beams produced by four-wave mixing in hot Rb vapor. A single phase modulator in either one of the twin beams reduces the two-mode squeezing signal, and we find that the modulations interfere nonlocally to modify the beam correlations. The nonlocal modulation of the beams can produce quantum correlations among frequency modes of the multimode fields.

Current attempts to test local position invariance (LPI) compare different clock transition rates with classically exchanged signals. We propose an experimental scheme for the quantum test of LPI: an internal atomic clock interferometer comprising two interfering clocks within one atom. We prepare the atom in a superposition of two clock states and one ground state, which evolves coherently along two quantum clock oscillations into stable internal Ramsey interference patterns. The interference pattern with the shared ground state shows a visibility modulation, which can be interpreted as the beating of the individual clock oscillations and a direct consequence of complementarity. Upon the interferometer experiencing a different gravitational potential, LPI predicts that both clock tick rates will change proportionally, while quantum complementarity indicates that the visibility modulation should modify accordingly. This change is deemed insignificant for the first period of visibility modulation but can be stacked up until the limit of the system coherence time. Since no splitting or recombining is involved, the system coherence time can be as long as the trap lifetime or the clock state lifetime. The required resolution to observe the visibility modulation is within reach of the state-of-art optical clocks' sensitivities. This experimental scheme is feasible in different scenarios, still or with speed, and may shed new light on studying the quantum effect of time and general relativity.

Moir\'e patterns are omnipresent. They are important for any overlapping periodic phenomenon, from vibrational and electromagnetic, to condensed matter. Here we show, both theoretically and via experimental simulations by ultracold atoms, that for one-dimensional finite-size periodic systems, moir\'e patterns give rise to anomalous features in both classical and quantum systems. In contrast to the standard moir\'e phenomenon, in which the pattern periodicity is a result of a beat-note between its constituents, we demonstrate moir\'e patterns formed from constituents with the same periodicity. Surprisingly, we observe, in addition, rigidity and singularities. We furthermore uncover universal properties in the frequency domain, which might serve as a novel probe of emitters. These one-dimensional effects could be relevant to a wide range of periodic phenomena.

The Stern-Gerlach effect, discovered a century ago, has become a paradigm of quantum mechanics. Surprisingly there has been little evidence that the original scheme with freely propagating atoms exposed to gradients from macroscopic magnets is a fully coherent quantum process. Specifically, no full-loop Stern-Gerlach interferometer has been realized with the scheme as envisioned decades ago. Furthermore, several theoretical studies have explained why such an interferometer is a formidable challenge. Here we provide a detailed account of the first full-loop Stern-Gerlach interferometer realization, based on highly accurate magnetic fields, originating from an atom chip, that ensure coherent operation within strict constraints described by previous theoretical analyses. Achieving this high level of control over magnetic gradients is expected to facilitate technological as well as fundamental applications, such as probing the interface of quantum mechanics and gravity. While the experimental realization described here is for a single atom, future challenges would benefit from utilizing macroscopic objects doped with a single spin. Specifically, we show that such an experiment is in principle feasible, opening the door to a new era of fundamental probes.

In this invited review in honor of 100 years since the Stern-Gerlach (SG) experiments, we describe a decade of SG interferometry on the atom chip. The SG effect has been a paradigm of quantum mechanics throughout the last century, but there has been surprisingly little evidence that the original scheme, with freely propagating atoms exposed to gradients from macroscopic magnets, is a fully coherent quantum process. Specifically, no full-loop SG interferometer (SGI) has been realized with the scheme as envisioned decades ago. Furthermore, several theoretical studies have explained why it is a formidable challenge. Here we provide a review of our SG experiments over the last decade. We describe several novel configurations such as that giving rise to the first SG spatial interference fringes, and the first full-loop SGI realization. These devices are based on highly accurate magnetic fields, originating from an atom chip, that ensure coherent operation within strict constraints described by previous theoretical analyses. Achieving this high level of control over magnetic gradients is expected to facilitate technological applications such as probing of surfaces and currents, as well as metrology. Fundamental applications include the probing of the foundations of quantum theory, gravity, and the interface of quantum mechanics and gravity. We end with an outlook describing possible future experiments., Comment: 20 pages, 10 figures, >100 references. Invited review from the September 2019 conference in Frankfurt celebrating the centennial of the Stern-Gerlach experiment

The geometric phase due to the evolution of the Hamiltonian is a central concept in quantum physics, and may become advantageous for quantum technology. In non-cyclic evolutions, a proposition relates the geometric phase to the area bounded by the phase-space trajectory and the shortest geodesic connecting its end points. The experimental verification of this geodesic rule proposition has remained elusive for more than three decades. Here, we report an unambiguous experimental confirmation of the geodesic rule for a non-cyclic geometric phase by means of a spatial SU(2) matter-wave interferometer, demonstrating, with high precision, the predicted phase sign change and pi jumps. We show the connection between our results and the Pancharatnam phase. Finally, we point out that the geodesic rule can be applied to obtain the red-shift in general relativity, enabling a completely new quantum tool to measure gravity.

Clocks play a special role at the interface of general relativity and quantum mechanics. We analyze a clock-interferometry thought experiment and go on to theoretically derive and experimentally test a complementarity relation for quantum clocks in the context of the gravitational time lag. We study this relation in detail and discuss its application to various types of quantum clocks., Comment: Supplementary Material included. Comments welcome!

The Stern-Gerlach (SG) effect, discovered almost a century ago, has become a paradigm of quantum mechanics. Surprisingly there is little evidence that the original scheme with freely propagating atoms exposed to gradients from macroscopic magnets is a fully coherent quantum process. Specifically, no high-visibility spatial interference pattern has been observed with such a scheme, and furthermore no full-loop SG interferometer has been realized with the scheme as envisioned decades ago. On the contrary, numerous theoretical studies explained why it is a near impossible endeavor. Here we demonstrate for the first time both a high-visibility spatial SG interference pattern and a full-loop SG interferometer, based on an accurate magnetic field, originating from an atom chip, that ensures coherent operation within strict constraints described by previous theoretical analyses. This also allows us to observe the gradual emergence of time-irreversibility as the splitting is increased. Finally, achieving this high level of control over magnetic gradients may facilitate technological applications such as large-momentum-transfer beam splitting for metrology with atom interferometry, ultra-sensitive probing of electron transport down to shot-noise and squeezed currents, as well as nuclear magnetic resonance and compact accelerators., Comment: 13 pages and 5 figures, and supplementary materials

Fundamentally, the dynamics of micro-macro transitions is instrumental to understanding the process of quantum-to-classical transitions; technologically, it can also facilitate the detection of the microscopic signals in quantum experiments via convenient detectors. Here, we demonstrate a scheme to characterize micro-macro transitions based on a four-wave mixing linear optical amplification process in a hot rubidium vapor. The linear optical amplifier provides a large optical gain of $10^7$ for injected single-photon-level pulses, enabling photon-number-resolving detection by average via non-single-photon counting detectors with a large dynamic range. The scheme exhibits strong dispersion which is sensitive to the input's change at the single-photon level, resulting in the group-velocity delay time scaling with $1/\sqrt{N}$, where $N$ is the average input photon number. The output probe and conjugate modes have different coefficients of this $1/\sqrt{N}$ scaling, indicating the coefficient can serve as an efficient parameter to characterize the specified micro-macro transitions. The demonstrated results are generally applicable for quantum detection and optical signal processing in light-atom interfaces. Furthermore, the present system is suitable for the study of relevant time-resolved dynamics of the quantum-to-classical transitions., Comment: 9 pages, 7 figures

We experimentally demonstrate a new interferometry paradigm: a self-interfering clock. We split a clock into two spatially separated wave packets, and observe an interference pattern with a stable phase showing that the splitting was coherent, i.e., the clock was in two places simultaneously. We then make the clock wave packets "tick" at different rates to simulate a proper time lag. The entanglement between the clock's time and its path yields "which path" information, which affects the visibility of the clock's self-interference. By contrast, in standard interferometry, time cannot yield "which path" information. As a clock we use an atom prepared in a superposition of two spin states. This first proof-of-principle experiment may have far-reaching implications for the study of time and general relativity and their impact on fundamental quantum effects such as decoherence and wave packet collapse., Comment: 22 pages, 8 figures

We show that it is possible to estimate the shape of an object by measuring only the fluctuations of a probing field, allowing us to expose the object to a minimal light intensity. This scheme, based on noise measurements through homodyne detection, is useful in the regime where the number of photons is low enough that direct detection with a photodiode is difficult but high enough such that photon counting is not an option. We generate a few-photon state of multi-spatial-mode vacuum-squeezed twin beams using four-wave mixing and direct one of these twin fields through a binary intensity mask whose shape is to be imaged. Exploiting either the classical fluctuations in a single beam or quantum correlations between the twin beams, we demonstrate that under some conditions quantum correlations can provide an enhancement in sensitivity when estimating the shape of the object.

We study the storage and retrieval of images in a hot atomic vapor using the gradient echo memory protocol. We demonstrate that this technique allows for the storage of multiple spatial modes. We study both spatial and temporal multiplexing by storing a sequence of two different images in the atomic vapor. The effect of atomic diffusion on the spatial resolution is discussed and characterized experimentally. For short storage time a normalized cross-correlation between a retrieved image and its input of 88 % is reported., Comment: 9 pages, to be published in Optics Express