Quantum noise limited and entanglement-assisted magnetometry

Roadmap on nanoscale magnetic resonance imaging

The field of nanoscale magnetic resonance imaging (NanoMRI) was started 30 years ago. It was motivated by the desire to image single molecules and molecular assemblies, such as proteins and virus particles, with near-atomic spatial resolution and on a length scale of 100 nm. Over the years, the NanoMRI field has also expanded to include the goal of useful high-resolution nuclear magnetic resonance (NMR) spectroscopy of molecules under ambient conditions, including samples up to the micron-scale. The realization of these goals requires the development of spin detection techniques that are many orders of magnitude more sensitive than conventional NMR and MRI, capable of detecting and controlling nanoscale ensembles of spins. Over the years, a number of different technical approaches to NanoMRI have emerged, each possessing a distinct set of capabilities for basic and applied areas of science. The goal of this roadmap article is to report the current state of the art in NanoMRI technologies, outline the areas where they are poised to have impact, identify the challenges that lie ahead, and propose methods to meet these challenges. This roadmap also shows how developments in NanoMRI techniques can lead to breakthroughs in emerging quantum science and technology applications.

An optically pumped magnetic gradiometer for the detection of human biomagnetism

We realise an intrinsic optically pumped magnetic gradiometer based on non-linear magneto-optical rotation. We show that our sensor can reach a gradiometric sensitivity of 18 fT cm - 1 Hz - 1 and can reject common mode homogeneous magnetic field noise with up to 30 dB attenuation. We demonstrate that our magnetic field gradiometer is sufficiently sensitive and resilient to be employed in biomagnetic applications. In particular, we are able to record the auditory evoked response of the human brain, and to perform real-time magnetocardiography in the presence of external magnetic field disturbances. Our gradiometer provides complementary capabilities in human biomagnetic sensing to optically pumped magnetometers, and opens new avenues in the detection of human biomagnetism.

Photon Pressure with an Effective Negative Mass Microwave Mode

Harmonic oscillators belong to the most fundamental concepts in physics and are central to many current research fields such as circuit QED, cavity optomechanics, and photon pressure systems. Here, we engineer a microwave mode in a superconducting LC circuit that mimics the dynamics of a negative mass oscillator, and couple it via photon pressure to a second low-frequency circuit. We demonstrate that the effective negative mass dynamics lead to an inversion of dynamical backaction and to sideband cooling of the low-frequency circuit by a blue-detuned pump field, which can be intuitively understood by the inverted energy ladder of a negative mass oscillator.

Optical circuit compactification for ultracold atoms

We develop a modular and compactified optical circuit for the generation of optical beams for cooling, imaging, and controlling ultracold atoms. One of the simplifications that is made in our circuit is to admix the repumping beams to each other optical beams in its dedicated single-mode fiber. We implement our design, characterize the output, and show that the optical power efficiency of the circuit is in the region of 97%, and after fiber coupling, the efficiencies are in the range of 62-85%. Given its compact design and controllable optical sources, this setup should be adaptable to a variety of quantum experiments based on ultracold gases.

Tomography of a Number-Resolving Detector by Reconstruction of an Atomic Many-Body Quantum State

The high-fidelity analysis of many-body quantum states of indistinguishable atoms requires the accurate counting of atoms. Here we report the tomographic reconstruction of an atom-number-resolving detector. The tomography is performed with an ultracold rubidium ensemble that is prepared in a coherent spin state by driving a Rabi coupling between the two hyperfine clock levels. The coupling is followed by counting the occupation number in one level. We characterize the fidelity of our detector and show that a negative-valued Wigner function is associated with it. Our results offer an exciting perspective for the high-fidelity reconstruction of entangled states and can be applied for a future demonstration of Heisenberg-limited atom interferometry.

Acoustic frequency atomic spin oscillator in the quantum regime

Quantum noise reduction and entanglement-enhanced sensing in the acoustic frequency range is an outstanding challenge relevant for a number of applications including magnetometry and broadband noise reduction in gravitational wave detectors. Here we experimentally demonstrate quantum behavior of a macroscopic atomic spin oscillator in the acoustic frequency range. Quantum back-action of the spin measurement, ponderomotive squeezing of light, and virtual spring softening are observed at oscillation frequencies down to the sub-kHz range. Quantum noise sources characteristic of spin oscillators operating in the near-DC frequency range are identified and means for their mitigation are presented.

Nuclear quadrupole resonance spectroscopy with a femtotesla diamond magnetometer

Radio frequency (RF) magnetometers based on nitrogen vacancy centers in diamond are predicted to offer femtotesla sensitivity, but previous experiments were limited to the picotesla level. We demonstrate a femtotesla RF magnetometer using a diamond membrane inserted between ferrite flux concentrators. The device provides ~300-fold amplitude enhancement for RF magnetic fields from 70 kHz to 3.6 MHz, and the sensitivity reaches ~70 fT√s at 0.35 MHz. The sensor detected the 3.6-MHz nuclear quadrupole resonance (NQR) of room-temperature sodium nitrite powder. The sensor's recovery time after an RF pulse is ~35 μs, limited by the excitation coil's ring-down time. The sodium-nitrite NQR frequency shifts with temperature as -1.00±0.02 kHz/K, the magnetization dephasing time is T2*=887±51 μs, and multipulse sequences extend the signal lifetime to 332±23 ms, all consistent with coil-based studies. Our results expand the sensitivity frontier of diamond magnetometers to the femtotesla range, with potential applications in security, medical imaging, and materials science.

Entanglement-Enhanced Magnetic Induction Tomography

Magnetic induction tomography (MIT) is a sensing protocol exploring conductive objects via their response to radio-frequency magnetic fields. MIT is used in nondestructive testing ranging from geophysics to medical applications. Atomic magnetometers, employed as MIT sensors, allow for significant improvement of the MIT sensitivity and for exploring its quantum limits. Here, we propose and verify a quantum-enhanced version of the atomic MIT by combining it with conditional spin squeezing and stroboscopic backaction evasion. We use this quantum enhancement to demonstrate sensitivity beyond the standard quantum limits of one-dimensional quantum MIT detecting a conductive sample.

Protection of Noise Squeezing in a Quantum Interferometer with Optimal Resource Allocation

Interferometers are crucial for precision measurements, including gravitational waves, laser ranging, radar, and imaging. The phase sensitivity, the core parameter, can be quantum-enhanced to break the standard quantum limit (SQL) using quantum states. However, quantum states are highly fragile and quickly degrade with losses. We design and demonstrate a quantum interferometer utilizing a beam splitter with a variable splitting ratio to protect the quantum resource against environmental impacts. The optimal phase sensitivity can reach the quantum Cramér-Rao bound of the system. This quantum interferometer can greatly reduce the quantum source requirements in quantum measurements. In theory, with a 66.6% loss rate, the sensitivity can break the SQL using only a 6.0 dB squeezed quantum resource with the current interferometer rather than a 24 dB squeezed quantum resource with a conventional squeezing-vacuum-injected Mach-Zehnder interferometer. In experiments, when using a 2.0 dB squeezed vacuum state, the sensitivity enhancement remains at ∼1.6  dB via optimizing the first splitting ratio when the loss rate changes from 0% to 90%, indicating that the quantum resource is excellently protected with the existence of losses in practical applications. This strategy could open a way to retain quantum advantages for quantum information processing and quantum precision measurement in lossy environments.

Quantum sensors for biomedical applications

Quantum sensors are finding their way from laboratories to the real world, as witnessed by the increasing number of start-ups in this field. The atomic length scale of quantum sensors and their coherence properties enable unprecedented spatial resolution and sensitivity. Biomedical applications could benefit from these quantum technologies, but it is often difficult to evaluate the potential impact of the techniques. This Review sheds light on these questions, presenting the status of quantum sensing applications and discussing their path towards commercialization. The focus is on two promising quantum sensing platforms: optically pumped atomic magnetometers, and nitrogen-vacancy centres in diamond. The broad spectrum of biomedical applications is highlighted by four case studies ranging from brain imaging to single-cell spectroscopy.

Different Configurations of Radio-Frequency Atomic Magnetometers-A Comparative Study

We comprehensively explore different optical configurations of a radio-frequency atomic magnetometer in the context of sensor miniaturisation. Similarities and differences in operation principles of the magnetometer arrangements are discussed. Through analysis of the radio-frequency and noise spectra, we demonstrate that all configurations provide the same level of atomic polarisation and signal-to-noise ratio, but the optimum performance is achieved for significantly different laser powers and frequencies. We conclude with possible strategies for system miniaturisation.

Unshielded portable optically pumped magnetometer for the remote detection of conductive objects using eddy current measurements

Electrically conductive objects can be detected using the principle of electromagnetic induction, where a primary oscillating magnetic field induces eddy currents in the object, which in turn produce a secondary magnetic field that can be measured with a magnetometer. We have developed a portable radio-frequency optically pumped magnetometer (RF OPM) working in unshielded conditions with sub-pT/Hz magnetic field sensitivity when used for the detection of small oscillating magnetic fields, setting a new benchmark for the sensitivity of a portable RF OPM in unshielded conditions. Using this OPM, we have detected the induced magnetic field from aluminum disks with diameters as small as 1.5 cm and with the disks being ∼25 cm from both the excitation coil and the magnetometer. When used for eddy current detection, our magnetometer achieves a sensitivity of a 2-6 pT/Hz. We have also detected a moving aluminum disk using our RF OPM and analyzed the magnetometer signals, which depend on the position of the disk, illustrating the potential of high sensitivity RF OPMs for remote sensing applications.

Moving-frame imaging of transiting cold atoms for precise long-range transport

Transporting cold atoms between interconnected vacuum chambers is an important technique for increasing the versatility of cold atom setups, particularly for those that couple atoms to photonic devices. In this report, we introduce a method where we are able to image the atoms at all points during transport via moving optical dipole trap. Cooled ⁸⁷Rb atoms are transported ∼50 cm into an auxiliary vacuum chamber while being monitored with a moving-frame imaging system for which in-situ characterization of the atom transport is demonstrated. Precise positioning of the atoms near photonic devices is also tested across several tapered fibers showing an axial positioning resolution of ∼450 μm.

Delta-Kick Squeezing

We explore the possibility to overcome the standard quantum limit (SQL) in a free-fall atom interferometer using a Bose-Einstein condensate (BEC) in either of the two relevant cases of Bragg or Raman scattering light pulses. The generation of entanglement in the BEC is dramatically enhanced by amplifying the atom-atom interactions via the rapid action of an external trap, focusing the matter waves to significantly increase the atomic densities during a preparation stage-a technique we refer to as delta-kick squeezing (DKS). The action of a second DKS operation at the end of the interferometry sequence allows one to implement a nonlinear readout scheme, making the sub-SQL sensitivity highly robust against imperfect atom counting detection. We predict more than 30 dB of sensitivity gain beyond the SQL for the variance, assuming realistic parameters and 10^{6}  atoms.

Momentum Entanglement for Atom Interferometry

Compared to light interferometers, the flux in cold-atom interferometers is low and the associated shot noise is large. Sensitivities beyond these limitations require the preparation of entangled atoms in different momentum modes. Here, we demonstrate a source of entangled atoms that is compatible with state-of-the-art interferometers. Entanglement is transferred from the spin degree of freedom of a Bose-Einstein condensate to well-separated momentum modes, witnessed by a squeezing parameter of -3.1(8)  dB. Entanglement-enhanced atom interferometers promise unprecedented sensitivities for quantum gradiometers or gravitational wave detectors.

Calibration of spin-light coupling by coherently induced Faraday rotation

Calibrating the strength of the light-matter interaction is an important experimental task in quantum information and quantum state engineering protocols. The strength of the off-resonant light-matter interaction in multi-atom spin oscillators can be characterized by the readout rate Γ_S. Here we introduce the method named Coherently Induced FAraday Rotation (CIFAR) for determining the readout rate. The method is suited for both continuous and pulsed readout of the spin oscillator, relying only on applying a known polarization modulation to the probe laser beam and detecting a known optical polarization component. Importantly, the method does not require changes to the optical and magnetic fields performing the state preparation and probing. The CIFAR signal is also independent of the probe beam photo-detection quantum efficiency, and allows direct extraction of other parameters of the interaction, such as the tensor coupling ζ_S, and the damping rate γ_S. We verify this method in the continuous wave regime, probing a strongly coupled spin oscillator prepared in a warm cesium atomic vapour.

Precise control of squeezing angle to generate 11 dB entangled state

The strength of the quantum correlations of a continuous-variable entangled state is determined by several relative phases in the preparation, transmission, and detection processes of entangled states. In this paper, we report the first experimental and theoretical demonstrations of the precision of relative phases associated with the strength of quadrature correlations. Based on the interrelations of the relative phases, three precisely phase-locking methodologies are established: ultralow RAM control loops for the lengths and relative phases stabilization of the DOPAs, difference DC locking for the relative phase between the two squeezed beams, and DC-AC joint locking for the relative phases in BHDs. The phase-locking loops ensure the total phase noise to be 9.7±0.32/11.1±0.36 mrad. Finally, all the relative phase deviations are controlled to be in the range of -35 to 35 mrad, which enhances the correlations of the amplitude and phase quadratures to -11.1 and -11.3 dB. The entanglement also exhibits a broadband squeezing bandwidth up to 100 MHz. This paves a valuable resource for experimental realization and applications in quantum information and precision measurement.

Waveform Estimation from Approximate Quantum Nondemolition Measurements

With the advent of gravitational wave detectors employing squeezed light, quantum waveform estimation-estimating a time-dependent signal by means of a quantum-mechanical probe-is of increasing importance. As is well known, backaction of quantum measurement limits the precision with which the waveform can be estimated, though these limits can, in principle, be overcome by "quantum nondemolition" (QND) measurement setups found in the literature. Strictly speaking, however, their implementation would require infinite energy, as their mathematical description involves Hamiltonians unbounded from below. This raises the question of how well one may approximate nondemolition setups with finite energy or finite-dimensional realizations. Here we consider a finite-dimensional waveform estimation setup based on the "quasi-ideal clock" and show that the estimation errors due to approximating the QND condition decrease slowly, as a power law, with increasing dimension. As a result, we find that approximating QND with this system requires large energy or dimensionality. We argue that this result can be expected to also hold for setups based on truncated oscillators or spin systems.

Noisy Quantum Metrology Enhanced by Continuous Nondemolition Measurement

We show that continuous quantum nondemolition (QND) measurement of an atomic ensemble is able to improve the precision of frequency estimation even in the presence of independent dephasing acting on each atom. We numerically simulate the dynamics of an ensemble with up to N=150 atoms initially prepared in a (classical) spin coherent state, and we show that, thanks to the spin squeezing dynamically generated by the measurement, the information obtainable from the continuous photocurrent scales superclassically with respect to the number of atoms N. We provide evidence that such superclassical scaling holds for different values of dephasing and monitoring efficiency. We moreover calculate the extra information obtainable via a final strong measurement on the conditional states generated during the dynamics and show that the corresponding ultimate limit is nearly achieved via a projective measurement of the spin-squeezed collective spin operator. We also briefly discuss the difference between our protocol and standard estimation schemes, where the state preparation time is neglected.

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.

Geometric Pathway to Scalable Quantum Sensing

Entangled resources enable quantum sensing that achieves Heisenberg scaling, a quadratic improvement on the standard quantum limit, but preparing large N spin entangled states is challenging in the presence of decoherence. We present a quantum control strategy using highly nonlinear geometric phase gates which can be used for generic state or unitary synthesis on the Dicke subspace with O(N) or O(N^{2}) gates, respectively. The method uses a dispersive coupling of the spins to a common bosonic mode and does not require addressability, special detunings, or interactions between the spins. By using amplitude amplification our control sequence for preparing states ideal for metrology can be significantly simplified to O(N^{5/4}) geometric phase gates with action angles O(1/N) that are more robust to mode decay. The geometrically closed path of the control operations ensures the gates are insensitive to the initial state of the mode and the sequence has built-in dynamical decoupling providing resilience to dephasing errors.

Dispersive microwave electrometry using Zeeman frequency modulation spectroscopy of electromagnetically induced transparency in Rydberg atoms

We herein developed and demonstrated a Zeeman frequency modulation scheme for improving the signal-to-noise ratio of microwave electric field measurement using Rydberg atoms. The spectra of the electromagnetically induced transparency (EIT) and Autler-Townes splitting of Rydberg atoms is frequency modulated by an alternating current magnetic field. The signal-to-noise ratio of the corresponding dispersive error signal is enhanced more than 10 times than that of the original spectrum. Furthermore, we show that the slope of the dispersive error signal near the resonance of the Rydberg EIT can be used to characterize the weak microwave electric field amplitudes. The more intuitive and simpler structure compared with other existing frequency modulation technologies greatly reduces the difficulties of experiments and experimental data analysis.

High-Precision Quantum-Enhanced Gravimetry with a Bose-Einstein Condensate

We show that the inherently large interatomic interactions of a Bose-Einstein condensate (BEC) can enhance the sensitivity of a high precision cold-atom gravimeter beyond the shot-noise limit (SNL). Through detailed numerical simulation, we demonstrate that our scheme produces spin-squeezed states with variances up to 14 dB below the SNL, and that absolute gravimetry measurement sensitivities between two and five times below the SNL are achievable with BECs between 10^{4} and 10^{6} in atom number. Our scheme is robust to phase diffusion, imperfect atom counting, and shot-to-shot variations in atom number and laser intensity. Our proposal is immediately achievable in current laboratories, since it needs only a small modification to existing state-of-the-art experiments and does not require additional guiding potentials or optical cavities.

Role of the probe beam in a radio-frequency atomic magnetometer

We explore the role of the linearly polarized probe beam in the radio-frequency atomic magnetometer. Two regimes of coupling of the linearly polarized light to the atomic ensemble, near and off-resonant, are demonstrated and discussed. Our studies cover three types of interaction between the atomic spin system and the linearly polarized beam, i.e., absorptive production of spin polarization, dispersive spin state readout, and nonlinear spin couplings. The work is performed with the outlook of optimization and simplification of magnetometer operation. Operation of a magnetometer with a single beam, generating and probing the atomic orientation, is presented. This single-beam configuration, which combines optical pumping, nonlinear spin couplings, and spin-exchange collisions, creates the option for a portable device with simple instrumentation.

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.

Long-Lived Entanglement Generation of Nuclear Spins Using Coherent Light

Nuclear spins of noble-gas atoms are exceptionally isolated from the environment and can maintain their quantum properties for hours at room temperature. Here we develop a mechanism for entangling two such distant macroscopic ensembles by using coherent light input. The interaction between the light and the noble-gas spins in each ensemble is mediated by spin-exchange collisions with alkali-metal spins, which are only virtually excited. The relevant conditions for experimental realizations with ^{3}He or ^{129}Xe are outlined.

Optical waveguiding by atomic entanglement in multilevel atom arrays

The optical properties of subwavelength arrays of atoms or other quantum emitters have attracted significant interest recently. For example, the strong constructive or destructive interference of emitted light enables arrays to function as nearly perfect mirrors, support topological edge states, and allow for exponentially better quantum memories. In these proposals, the assumed atomic structure was simple, consisting of a unique electronic ground state. Within linear optics, the system is then equivalent to a periodic array of classical dielectric particles, whose periodicity supports the emergence of guided modes. However, it has not been known whether such phenomena persist in the presence of hyperfine structure, as exhibited by most quantum emitters. Here, we show that waveguiding can arise from rich atomic entanglement as a quantum many-body effect and elucidate the necessary conditions. Our work represents a significant step forward in understanding collective effects in arrays of atoms with realistic electronic structure.

Demonstrating Quantum Coherence and Metrology that is Resilient to Transversal Noise

Quantum systems can be exploited for disruptive technologies but in practice quantum features are fragile due to noisy environments. Quantum coherence, a fundamental such feature, is a basis-dependent property that is known to exhibit a resilience to certain types of Markovian noise. Yet, it is still unclear whether this resilience can be relevant in practical tasks. Here, we experimentally investigate the resilient effect of quantum coherence in a photonic Greenberger-Horne-Zeilinger state under Markovian bit-flip noise, and explore its applications in a noisy metrology scenario. In particular, using up to six-qubit probes, we demonstrate that the standard quantum limit can be outperformed under a transversal noise strength of approximately equal magnitude to the signal, providing experimental evidence of metrological advantage even in the presence of uncorrelated Markovian noise. This work highlights the important role of passive control in noisy quantum hardware, which can act as a low-overhead complement to more traditional approaches such as quantum error correction, thus impacting on the deployment of quantum technologies in real-world settings.

Motional Fock states for quantum-enhanced amplitude and phase measurements with trapped ions

The quantum noise of the vacuum limits the achievable sensitivity of quantum sensors. In non-classical measurement schemes the noise can be reduced to overcome this limitation. However, schemes based on squeezed or Schrödinger cat states require alignment of the relative phase between the measured interaction and the non-classical quantum state. Here we present two measurement schemes on a trapped ion prepared in a motional Fock state for displacement and frequency metrology that are insensitive to this phase. The achieved statistical uncertainty is below the standard quantum limit set by quantum vacuum fluctuations, enabling applications in spectroscopy and mass measurements.

Quantum metrology allows surpassing the standard quantum limit, but methods relying on squeezing require to know the orientation of the squeezed quadrature with respect to the signal. Here, instead, the authors propose a phase-insensitive Fock-state-based protocol, and demonstrate it using trapped ions.

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.

Fabrication and characterization of super-polished wedged borosilicate nano-cells

We report on the fabrication of an all-glass vapor cell with a thickness varying linearly between (exactly) 0 and ∼1  μm. The cell is made in Borofloat glass that allows state-of-the-art super polish roughness, a full optical bonding assembling and easy filling with alkali vapors. We detail the challenging manufacture steps and present experimental spectra resulting from fluorescence and transmission spectroscopy of the cesium D1 line. The very small surface roughness of 1 Å rms is promising to investigate the atom-surface interaction or to minimize parasite stray light.

Imaging the Local Charge Environment of Nitrogen-Vacancy Centers in Diamond

Characterizing the local internal environment surrounding solid-state spin defects is crucial to harnessing them as nanoscale sensors of external fields. This is especially germane to the case of defect ensembles which can exhibit a complex interplay between interactions, internal fields, and lattice strain. Working with the nitrogen-vacancy (NV) center in diamond, we demonstrate that local electric fields dominate the magnetic resonance behavior of NV ensembles at a low magnetic field. We introduce a simple microscopic model that quantitatively captures the observed spectra for samples with NV concentrations spanning more than two orders of magnitude. Motivated by this understanding, we propose and implement a novel method for the nanoscale localization of individual charges within the diamond lattice; our approach relies upon the fact that the charge induces a NV dark state which depends on the electric field orientation.

Magnetocardiography on an isolated animal heart with a room-temperature optically pumped magnetometer

Optically pumped magnetometers are becoming a promising alternative to cryogenically-cooled superconducting magnetometers for detecting and imaging biomagnetic fields. Magnetic field detection is a completely non-invasive method, which allows one to study the function of excitable human organs with a sensor placed outside the human body. For instance, magnetometers can be used to detect brain activity or to study the activity of the heart. We have developed a highly sensitive miniature optically pumped magnetometer based on cesium atomic vapor kept in a paraffin-coated glass container. The magnetometer is optimized for detection of biological signals and has high temporal and spatial resolution. It is operated at room- or human body temperature and can be placed in contact with or at a mm-distance from a biological object. With this magnetometer, we detected the heartbeat of an isolated guinea-pig heart, which is an animal widely used in biomedical studies. In our recordings of the magnetocardiogram, we can detect the P-wave, QRS-complex and T-wave associated with the cardiac cycle in real time. We also demonstrate that our device is capable of measuring the cardiac electrographic intervals, such as the RR- and QT-interval, and detecting drug-induced prolongation of the QT-interval, which is important for medical diagnostics.

Relativistic Measurement Backaction in the Quantum Dirac Oscillator

An elegant method to circumvent quantum measurement backaction is the use of quantum mechanics free subsystems (QMFS), with one approach involving the use of two oscillators with effective masses of opposite signs. Since negative energies, and hence masses, are a characteristic of relativistic systems a natural question is to what extent QMFS can be realized in this context. Using the example of a one-dimensional Dirac oscillator we investigate conditions under which this can be achieved, and identify Zitterbewegung or virtual pair creation as the physical mechanism that fundamentally limits the feasibility of the scheme. We propose a tabletop implementation of a Dirac oscillator system based on a spin-orbit coupled ultracold atomic sample that allows for a direct observation of the corresponding analog of virtual pair creation on quantum measurement backaction.

Unconditional Steady-State Entanglement in Macroscopic Hybrid Systems by Coherent Noise Cancellation

The generation of entanglement between disparate physical objects is a key ingredient in the field of quantum technologies, since they can have different functionalities in a quantum network. Here we propose and analyze a generic approach to steady-state entanglement generation between two oscillators with different temperatures and decoherence properties coupled in cascade to a common unidirectional light field. The scheme is based on a combination of coherent noise cancellation and dynamical cooling techniques for two oscillators with effective masses of opposite signs, such as quasispin and motional degrees of freedom, respectively. The interference effect provided by the cascaded setup can be tuned to implement additional noise cancellation leading to improved entanglement even in the presence of a hot thermal environment. The unconditional entanglement generation is advantageous since it provides a ready-to-use quantum resource. Remarkably, by comparing to the conditional entanglement achievable in the dynamically stable regime, we find our unconditional scheme to deliver a virtually identical performance when operated optimally.

Overcoming the Standard Quantum Limit in Gravitational Wave Detectors Using Spin Systems with a Negative Effective Mass

Quantum backaction (QBA) of a measurement limits the precision of observation of the motion of a free mass. This profound effect, dubbed the "Heisenberg microscope" in the early days of quantum mechanics, leads to the standard quantum limit (SQL) stemming from the balance between the measurement sensitivity and the QBA. We consider the measurement of motion of a free mass performed in a quantum reference frame with an effective negative mass which is not limited by QBA. As a result, the disturbance on the motion of a free mass can be measured beyond the SQL. QBA-limited detection of motion for a free mass is extremely challenging, but there are devices where this effect is expected to play an essential role, namely, gravitational wave detectors (GWDs) such as LIGO and Virgo. Recent reports on the observations of gravitational waves have opened new horizons in cosmology and astrophysics. We present a general idea and a detailed numerical analysis for QBA-evading measurement of the gravitational wave effect on the GWD mirrors, which can be considered free masses under relevant conditions. The measurement is performed by two entangled beams of light, probing the GWD and an auxiliary atomic spin ensemble, respectively. The latter plays the role of a free negative mass. We show that under realistic conditions the sensitivity of the GWD in m/sqrt[Hz] can be increased by 6 dB over the entire frequency band of interest.

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.

Quantum metrology with quantum-chaotic sensors

Quantum metrology promises high-precision measurements of classical parameters with far reaching implications for science and technology. So far, research has concentrated almost exclusively on quantum-enhancements in integrable systems, such as precessing spins or harmonic oscillators prepared in non-classical states. Here we show that large benefits can be drawn from rendering integrable quantum sensors chaotic, both in terms of achievable sensitivity as well as robustness to noise, while avoiding the challenge of preparing and protecting large-scale entanglement. We apply the method to spin-precession magnetometry and show in particular that the sensitivity of state-of-the-art magnetometers can be further enhanced by subjecting the spin-precession to non-linear kicks that renders the dynamics chaotic.

Enhanced sensing using quantum systems generally relies on preparation of highly non-classical states, such as entangled or squeezed states. Here the authors propose a different approach, based on hypersensitivity of quantum-chaotic dynamics with respect to parameters of the system.

Single-Atom Transistor as a Precise Magnetic Field Sensor

Feshbach resonances, which allow for tuning the interactions of ultracold atoms with an external magnetic field, have been widely used to control the properties of quantum gases. We propose a scheme for using scattering resonances as a probe for external fields, showing that by carefully tuning the parameters it is possible to reach a 10^{-5}  G (or nT) level of precision with a single pair of atoms. We show that, for our collisional setup, it is possible to saturate the quantum precision bound with a simple measurement protocol.

In situ magnetometry for experiments with atomic quantum gases

Precise control of magnetic fields is a frequent challenge encountered in experiments with atomic quantum gases. Here we present a simple method for performing in situ monitoring of magnetic fields that can readily be implemented in any quantum-gas apparatus in which a dedicated field-stabilization approach is not feasible. The method, which works by sampling several Rabi resonances between magnetically field sensitive internal states that are not otherwise used in a given experiment, can be integrated with standard measurement sequences at arbitrary fields. For a condensate of ⁸⁷Rb atoms, we demonstrate the reconstruction of Gauss-level bias fields with an accuracy of tens of microgauss and with millisecond time resolution. We test the performance of the method using measurements of slow resonant Rabi oscillations on a magnetic-field sensitive transition and give an example for its use in experiments with state-selective optical potentials.

Optimization of photon storage fidelity in ordered atomic arrays

A major application for atomic ensembles consists of a quantum memory for light, in which an optical state can be reversibly converted to a collective atomic excitation on demand. There exists a well-known fundamental bound on the storage error, when the ensemble is describable by a continuous medium governed by the Maxwell-Bloch equations. However, these equations are semi-phenomenological, as they treat emission of the atoms into other directions other than the mode of interest as being independent. On the other hand, in systems such as dense, ordered atomic arrays, atoms interact with each other strongly and spatial interference of the emitted light might be exploited to suppress emission into unwanted directions, thereby enabling improved error bounds. Here, we develop a general formalism that fully accounts for spatial interference, and which finds the maximum storage efficiency for a single photon with known spatial input mode into a collection of atoms with discrete, known positions. As an example, we apply this technique to study a finite two-dimensional square array of atoms. We show that such a system enables a storage error that scales with atom number N a like ∼ ( log N a ) 2 ∕ N a 2 , and that, remarkably, an array of just 4 × 4 atoms in principle allows for an error of less than 1%, which is comparable to a disordered ensemble with an optical depth of around 600.

Light-matter quantum interferometry with homodyne detection

We investigated the estimation of an unknown Gaussian process (containing displacement, squeezing and phase-shift) applied to a matter system. The state of the matter system is not directly measured; instead, we measure an optical mode which interacts with the system. We propose an interferometric setup exploiting a beam-splitter-type of light-matter interaction with homodyne detectors and two methods of estimation. We demonstrate the superiority of the interferometric setup over alternative non-interferometric schemes. Importantly, we show that even limited coupling strength and a noisy matter system are sufficient for very good estimation. Our work opens the way to many future investigations of light-matter interferometry for experimental platforms in quantum metrology of matter systems.

Atom-Based Sensing of Weak Radio Frequency Electric Fields Using Homodyne Readout

We utilize a homodyne detection technique to achieve a new sensitivity limit for atom-based, absolute radio-frequency electric field sensing of 5 μV cm⁻¹ Hz^−1/2. A Mach-Zehnder interferometer is used for the homodyne detection. With the increased sensitivity, we investigate the dominant dephasing mechanisms that affect the performance of the sensor. In particular, we present data on power broadening, collisional broadening and transit time broadening. Our results are compared to density matrix calculations. We show that photon shot noise in the signal readout is currently a limiting factor. We suggest that new approaches with superior readout with respect to photon shot noise are needed to increase the sensitivity further.

Prediction and real-time compensation of qubit decoherence via machine learning

The wide-ranging adoption of quantum technologies requires practical, high-performance advances in our ability to maintain quantum coherence while facing the challenge of state collapse under measurement. Here we use techniques from control theory and machine learning to predict the future evolution of a qubit's state; we deploy this information to suppress stochastic, semiclassical decoherence, even when access to measurements is limited. First, we implement a time-division multiplexed approach, interleaving measurement periods with periods of unsupervised but stabilised operation during which qubits are available, for example, in quantum information experiments. Second, we employ predictive feedback during sequential but time delayed measurements to reduce the Dick effect as encountered in passive frequency standards. Both experiments demonstrate significant improvements in qubit-phase stability over ‘traditional' measurement-based feedback approaches by exploiting time domain correlations in the noise processes. This technique requires no additional hardware and is applicable to all two-level quantum systems where projective measurements are possible.

Control engineering techniques are promising for realizing stable quantum systems to counter their extreme fragility. Here the authors use techniques from machine learning to enable real-time feedback suppression of decoherence in a trapped ion qubit by predicting its future stochastic evolution.

Pulsed operation of a miniature scalar optically pumped magnetometer

A scalar magnetic field sensor based on a millimeter-size ⁸⁷Rb vapor cell is described. The magnetometer uses nearly copropagating pump and probe laser beams, amplitude modulation of the pump beam, and detection through monitoring the polarization rotation of the detuned probe beam. The circularly polarized pump laser resonantly drives a spin precession in the alkali atoms at the Larmor frequency. A modulation signal on the probe laser polarization is detected with a lock-in amplifier. Since the Larmor precession is driven all-optically, potential cross talk between sensors is minimized. And since the pump light is turned off during most of the precession cycle, large offsets of the resonance, typically present in a single-beam Bell-Bloom scheme, are avoided. At the same time, relatively high sensitivities can be reached even in millimeter-size vapor cells: The magnetometer achieves a sensitivity of 1 pT/Hz^1/2 in a sensitive volume of 16 mm³, limited by environmental noise. When a gradiometer configuration is used to cancel the environmental noise, the magnetometer sensitivity reaches 300 fT/Hz^1/2. We systematically study the dependence of the magnetometer performance on the optical duty cycles of the pump light and find that better performance is achieved with shorter duty cycles, with the highest values measured at 1.25% duty cycle.