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Dynamic critical fluctuations in magnetic materials encode important information about magnetic ordering in the associated critical exponents. Using nitrogen-vacancy centers in diamond, we implement $T_2$ (spin-decoherence) noise magnetometry to study critical dynamics in a 2D Van der Waals magnet CrSBr. By analyzing NV decoherence on time scales approaching the characteristic correlation time $\tau_c$ of critical fluctuations, we extract the critical exponent $\nu$ for the correlation length. Our result deviates from the Ising prediction and highlights the role of long-range dipolar interactions in 2D CrSBr. Furthermore, analyzing the divergence of the correlation length suggests the possibility of 2D-XY criticality in CrSBr in a temperature window near $T_C$ where static magnetic domains are absent. Our work provides a first demonstration of $T_2$ noise magnetometry to quantitatively analyze critical scaling behavior in 2D materials., Comment: 19 pages main text, 4 main text figures, 26 pages Supplementary Materials, 13 Supplementary figures

The negatively charged boron vacancy ($\mathrm{V}_{\mathrm{B}}^-$) in hexagonal boron nitride (hBN) has garnered significant attention among defects in two-dimensional materials. This owes, in part, to its deterministic generation, well-characterized atomic structure, and optical polarizability at room temperature. We investigate the latter through extensive measurements probing both the ground and excited state polarization dynamics. We develop a semiclassical model based on these measurements that predicts a near-unity degree of spin polarization, surpassing other solid-state spin defects under ambient conditions. Building upon our model, we include the presence of nuclear spin degrees of freedom adjacent to the $\mathrm{V}_{\mathrm{B}}^-$ and perform a comprehensive set of Lindbladian numerics to investigate the hyperfine-induced polarization of the nuclear spins. Our simulations predict a number of important features that emerge as a function of magnetic field which are borne out by experiment.

Systems of spins with strong dipolar interactions and controlled dimensionality enable new explorations in quantum sensing and simulation. In this work, we investigate the creation of strong dipolar interactions in a two-dimensional ensemble of nitrogen-vacancy (NV) centers generated via plasma-enhanced chemical vapor deposition (PECVD) on (111)-oriented diamond substrates. We find that diamond growth on the (111) plane yields high incorporation of spins, both nitrogen and NV centers, where the density of the latter is tunable via the miscut of the diamond substrate. Our process allows us to form dense, preferentially aligned, 2D NV ensembles with volume-normalized AC sensitivity down to $\eta_{AC}$ = 810 pT um$^{3/2}$ Hz$^{-1/2}$. Furthermore, we show that (111) affords maximally positive dipolar interactions amongst a 2D NV ensemble, which is crucial for leveraging dipolar-driven entanglement schemes and exploring new interacting spin physics.

By directly altering microscopic interactions, pressure provides a powerful tuning knob for the exploration of condensed phases and geophysical phenomena. The megabar regime represents an exciting frontier, where recent discoveries include novel high-temperature superconductors, as well as structural and valence phase transitions. However, at such high pressures, many conventional measurement techniques fail. Here, we demonstrate the ability to perform local magnetometry inside of a diamond anvil cell with sub-micron spatial resolution at megabar pressures. Our approach utilizes a shallow layer of Nitrogen-Vacancy (NV) color centers implanted directly within the anvil; crucially, we choose a crystal cut compatible with the intrinsic symmetries of the NV center to enable functionality at megabar pressures. We apply our technique to characterize a recently discovered hydride superconductor, CeH$_9$. By performing simultaneous magnetometry and electrical transport measurements, we observe the dual signatures of superconductivity: local diamagnetism characteristic of the Meissner effect and a sharp drop of the resistance to near zero. By locally mapping the Meissner effect and flux trapping, we directly image the geometry of superconducting regions, revealing significant inhomogeneities at the micron scale. Our work brings quantum sensing to the megabar frontier and enables the closed loop optimization of superhydride materials synthesis.

Spin squeezing is a form of entanglement that reshapes the quantum projection noise to improve measurement precision. Here, we provide numerical and analytic evidence for the following conjecture: any Hamiltonian exhibiting finite temperature, easy-plane ferromagnetism can be used to generate scalable spin squeezing, thereby enabling quantum-enhanced sensing. Our conjecture is guided by a connection between the quantum Fisher information of pure states and the spontaneous breaking of a continuous symmetry. We demonstrate that spin-squeezing exhibits a phase diagram with a sharp transition between scalable squeezing and non-squeezing. This transition coincides with the equilibrium phase boundary for XY order at a finite temperature. In the scalable squeezing phase, we predict a sensitivity scaling that lies in between the standard quantum limit and the scaling achieved in all-to-all coupled one-axis twisting models. A corollary of our conjecture is that short-ranged versions of two-axis twisting cannot yield scalable metrological gain. Our results provide insights into the landscape of Hamiltonians that can be used to generate metrologically useful quantum states., Comment: 6 pages, 3 figures + 24 pages, 13 figures

Systems of spins engineered with tunable density and reduced dimensionality enable a number of advancements in quantum sensing and simulation. Defects in diamond, such as nitrogen-vacancy (NV) centers and substitutional nitrogen (P1 centers), are particularly promising solid-state platforms to explore. However, the ability to controllably create coherent, two-dimensional spin systems and characterize their properties, such as density, depth confinement, and coherence is an outstanding materials challenge. We present a refined approach to engineer dense ($\gtrsim$1 ppm$\cdot$nm), 2D nitrogen and NV layers in diamond using delta-doping during plasma-enhanced chemical vapor deposition (PECVD) epitaxial growth. We employ both traditional materials techniques, e.g. secondary ion mass spectrometry (SIMS), alongside NV spin decoherence-based measurements to characterize the density and dimensionality of the P1 and NV layers. We find P1 densities of 5-10 ppm$\cdot$nm, NV densities between 1 and 3.5 ppm$\cdot$nm tuned via electron irradiation dosage, and depth confinement of the spin layer down to 1.6 nm. We also observe high (up to 0.74) ratios of P1 to NV centers and reproducibly long NV coherence times, dominated by dipolar interactions with the engineered P1 and NV spin baths.

Interactions govern the flow of information and the formation of correlations in quantum systems, dictating the phases of matter found in nature and the forms of entanglement generated in the laboratory. Typical interactions decay with distance and thus produce a network of connectivity governed by geometry, e.g., by the crystalline structure of a material or the trapping sites of atoms in a quantum simulator. However, many envisioned applications in quantum simulation and computation require richer coupling graphs including nonlocal interactions, which notably feature in mappings of hard optimization problems onto frustrated spin systems and in models of information scrambling in black holes. Here, we report on the realization of programmable nonlocal interactions in an array of atomic ensembles within an optical cavity, where photons carry information between distant atomic spins. By programming the distance-dependence of interactions, we access effective geometries where the dimensionality, topology, and metric are entirely distinct from the physical arrangement of atoms. As examples, we engineer an antiferromagnetic triangular ladder, a Moebius strip with sign-changing interactions, and a treelike geometry inspired by concepts of quantum gravity. The tree graph constitutes a toy model of holographic duality, where the quantum system may be viewed as lying on the boundary of a higher-dimensional geometry that emerges from measured spin correlations. Our work opens broader prospects for simulating frustrated magnets and topological phases, investigating quantum optimization algorithms, and engineering new entangled resource states for sensing and computation.

The most direct approach for characterizing the quantum dynamics of a strongly-interacting system is to measure the time-evolution of its full many-body state. Despite the conceptual simplicity of this approach, it quickly becomes intractable as the system size grows. An alternate framework is to think of the many-body dynamics as generating noise, which can be measured by the decoherence of a probe qubit. Our work centers on the following question: What can the decoherence dynamics of such a probe tell us about the many-body system? In particular, we utilize optically addressable probe spins to experimentally characterize both static and dynamical properties of strongly-interacting magnetic dipoles. Our experimental platform consists of two types of spin defects in diamond: nitrogen-vacancy (NV) color centers (probe spins) and substitutional nitrogen impurities (many-body system). We demonstrate that signatures of the many-body system's dimensionality, dynamics, and disorder are naturally encoded in the functional form of the NV's decoherence profile. Leveraging these insights, we directly characterize the two-dimensional nature of a nitrogen delta-doped diamond sample. In addition, we explore two distinct facets of the many-body dynamics: First, we address a persistent debate about the microscopic nature of spin dynamics in strongly-interacting dipolar systems. Second, we demonstrate direct control over the correlation time of the many-body system. Finally, we demonstrate polarization exchange between NV and P1 centers, opening the door to quantum sensing and simulation using two-dimensional spin-polarized ensembles., Comment: 30 + 18 + 7 pages; 4 + 7 figures

Numerous conceptually important quantum algorithms rely on a black-box device known as an oracle, which is typically difficult to construct without knowing the answer to the problem that the algorithm is intended to solve. A notable example is Grover's search algorithm. Here we propose a Grover search for solutions to a class of NP-complete decision problems known as subset sum problems, including the special case of number partitioning. Each problem instance is encoded in the couplings of a set of qubits to a central spin or boson, which enables a realization of the oracle without knowledge of the solution. The algorithm provides a quantum speedup across a known phase transition in the computational complexity of the partition problem, and we identify signatures of the phase transition in the simulated performance. Whereas the naive implementation of our algorithm requires a spectral resolution that scales exponentially with system size for NP-complete problems, we also present a recursive algorithm that enables scalability. We propose and analyze implementation schemes with cold atoms, including Rydberg-atom and cavity-QED platforms., Comment: 23 pages, 13 figures, typos corrected, edits for clarity

Using an ensemble of atoms in an optical cavity, we engineer a family of nonlocal Heisenberg Hamiltonians with continuously tunable anisotropy of the spin-spin couplings. We thus gain access to a rich phase diagram, including a paramagnetic-to-ferromagnetic Ising phase transition that manifests as a diverging magnetic susceptibility at the critical point. The susceptibility displays a symmetry between Ising interactions and XY (spin-exchange) interactions of the opposite sign, which is indicative of the spatially extended atomic system behaving as a single collective spin. Images of the magnetization dynamics show that spin-exchange interactions protect the coherence of the collective spin, even against inhomogeneous fields that completely dephase the non-interacting and Ising systems. Our results underscore prospects for harnessing spin-exchange interactions to enhance the robustness of spin squeezing protocols.

Entanglement entropy is an essential metric for characterizing quantum many-body systems, but its numerical evaluation for neural network representations of quantum states has so far been inefficient and demonstrated only for the restricted Boltzmann machine architecture. Here, we estimate generalized Renyi entropies of autoregressive neural quantum states with up to N=256 spins using quantum Monte Carlo methods. A naive "direct sampling" approach performs well for low-order Renyi entropies but fails for larger orders when benchmarked on a 1D Heisenberg model. We therefore propose an improved "conditional sampling" method exploiting the autoregressive structure of the network ansatz, which outperforms direct sampling and facilitates calculations of higher-order Renyi entropies in both 1D and 2D Heisenberg models. Access to higher-order Renyi entropies allows for an approximation of the von Neumann entropy as well as extraction of the single copy entanglement. Both methods elucidate the potential of neural network quantum states in quantum Monte Carlo studies of entanglement entropy for many-body systems.

Operator scrambling denotes the evolution of a simple operator into a complicated one (in the Heisenberg picture), which characterizes quantum chaos in many-body systems. More specifically, a simple operator evolves into a linear superposition of many operators, most of which are many-body operators supported on a region of size much larger than $1$. In general, an operator does not have a definite size but is characterized by a probability distribution of size. The operator size is related to out-of-time-order correlation functions, but these are generically difficult to obtain from experimental observables. In this paper we show that the operator size distribution can be measured in quantum quench experiments. In a quantum spin system, we propose to prepare an ensemble of initial states which are direct product states of random pure states of each spin qudit, and measure a simple physical observable (such as a particular component of spin) at later time $t$. The initial state dependence of the expectation value measures a particular component of the operator size distribution. Furthermore, many other features of the operator size distribution can be measured by analyzing the same data, such as the support of the operator in space., Comment: 19 pages, 8 figures

We propose an experimentally realizable quantum spin model that exhibits fast scrambling, based on non-local interactions which couple sites whose separation is a power of 2. By controlling the relative strengths of deterministic, non-random couplings, we can continuously tune from the linear geometry of a nearest-neighbor spin chain to an ultrametric geometry in which the effective distance between spins is governed by their positions on a tree graph. The transition in geometry can be observed in quench dynamics, and is furthermore manifest in calculations of the entanglement entropy. Between the linear and treelike regimes, we find a peak in entanglement and exponentially fast spreading of quantum information across the system. Our proposed implementation, harnessing photon-mediated interactions among cold atoms in an optical cavity, offers a test case for experimentally observing the emergent geometry of a quantum many-body system., Comment: 6 pages, 4 figures (plus 7-page supplement with 4 figures)

We introduce NetKet, a comprehensive open source framework for the study of many-body quantum systems using machine learning techniques. The framework is built around a general and flexible implementation of neural-network quantum states, which are used as a variational ansatz for quantum wave functions. NetKet provides algorithms for several key tasks in quantum many-body physics and quantum technology, namely quantum state tomography, supervised learning from wave-function data, and ground state searches for a wide range of customizable lattice models. Our aim is to provide a common platform for open research and to stimulate the collaborative development of computational methods at the interface of machine learning and many-body physics.

We report direct observations of photon-mediated spin-exchange interactions in an atomic ensemble. Interactions extending over a distance of 500 microns are generated within a cloud of cold rubidium atoms coupled to a single mode of light in an optical resonator. We characterize the system via quench dynamics and imaging of the local magnetization, verifying the coherence of the interactions and demonstrating optical control of their strength and sign. Furthermore, by initializing the spin-1 system in the mF = 0 Zeeman state, we observe correlated pair creation in the mF = +/- 1 states, a process analogous to spontaneous parametric down-conversion and to spin mixing in Bose-Einstein condensates. Our work opens new opportunities in quantum simulation with long-range interactions and in entanglement-enhanced metrology.

We propose a robust scheme for generating macroscopic superposition states of spin or motion with the aid of a single photon. Shaping the wave packet of the photon enables high-fidelity preparation of non-classical states of matter even in the presence of photon loss. Success is heralded by photodetection, enabling the scheme to be implemented with a weak coherent field. We analyze applications to preparing Schr\"{o}dinger cat states of a collective atomic spin or of a mechanical oscillator coupled to an optical resonator. The method generalizes to preparing arbitrary superpositions of coherent states, enabling full quantum control. We illustrate this versatility by showing how to prepare Dicke or Fock states, as well as superpositions in the Dicke or Fock basis.

The strong light-matter coupling attainable in optical cavities enables the generation of highly squeezed states of atomic ensembles. It was shown in [Phys. Rev. A 66, 022314 (2002)] how an effective one-axis twisting Hamiltonian can be realized in a cavity setup. Here, we extend this work and show how an effective two-axis twisting Hamiltonian can be realized in a similar cavity setup. We compare the two schemes in order to characterize their advantages. In the absence of decoherence, the two-axis Hamiltonian leads to more squeezing than the one-axis Hamiltonian. If limited by decoherence from spontaneous emission and cavity decay, we find roughly the same level of squeezing for the two schemes scaling as (NC)^(1/2) where C is the single atom cooperativity and N is the total number of atoms. When compared to an ideal squeezing operation, we find that for specific initial states, a dissipative version of the one-axis scheme attains higher fidelity than the unitary one-axis scheme or the two-axis scheme. However, the unitary one-axis and two-axis schemes perform better for general initial states., Comment: 13 pages, 6 figures

We propose an approach to quantum phase estimation that can attain precision near the Heisenberg limit without requiring single-particle-resolved state detection. We show that the "one-axis twisting" interaction, well known for generating spin squeezing in atomic ensembles, can also amplify the output signal of an entanglement-enhanced interferometer to facilitate readout. Applying this interaction-based readout to oversqueezed, non-Gaussian states yields a Heisenberg scaling in phase sensitivity, which persists in the presence of detection noise as large as the quantum projection noise of an unentangled ensemble. Even in dissipative implementations -- e.g., employing light-mediated interactions in an optical cavity or Rydberg dressing -- the method significantly relaxes the detection resolution required for spectroscopy beyond the standard quantum limit., Comment: 5 + 4 pages (main text + supplement), 4 figures; in press at PRL

We present a scheme for locking optical cavities with arbitrary detuning many line widths from resonance using an electro-optic modulator that can provide arbitrary ratios of amplitude to phase modulation. We demonstrate our scheme on a Fabry-Perot cavity, and show that a well-behaved linear error signal can be obtained by demodulating the reflected light from a cavity that is detuned by several line widths., Comment: 5 pages, 4 figures

Clinical Question: For adult patients with cognitive linguistic impairment because of traumatic brain injury (TBI), is the use of internal and/or external memory strategies beneficial to memory outcomes compared to alternative or no treatment? Method: Systematic Review. Study Sources: Comprehensive EBSCOhost database search (i.e., ERIC, Academic Search Complete, Psychology and Behavioral Sciences Collection, Education Source), speechBITE, Cochrane Library, PubMed, PsycINFO, PsycBITE, and the ASHA Practice Portal. Search Terms: traumatic brain injury OR TBI AND intervention OR treatment OR therapy, cognit*, memory, internal strateg* OR external strateg*. Number of Included Studies: 11. Primary Results: The use of internal and external strategies to facilitate positive memory outcomes for TBI patients is supported. A variety of internal and external strategies have been studied with consistent positive results. Conclusion: Both internal and external strategies should be considered for use as compensatory techniques for memory impairments in adults with traumatic brain injuries. The evidence presented in this paper explores a variety of internal and external strategies, and outcomes have revealed an overall positive effect. These strategies have important implications for clinical decision-making for improving patient quality of life after TBI.

In this paper we give a complete characterization of linear, quadratic, and geometric Legendre multiplier sequences. We also prove that all Legendre multiplier sequences must be Hermite multiplier sequences, and describe the relationship between the Legendre and generalized Laguerre multiplier sequences. We conclude with a list of open questions for further research., Comment: 12 pages, 5 figures