Summary form only given. Continuous monitoring or tracking of a quantum system is essential to high-sensitivity measurement of time-varying quantities. We estimate spin precession of a magnetically-sensitive atomic ensemble by measuring spin projectors Fαat different times. Such projectors do not commute, and thus quantum measurement back-action (QMBA) necessarily enters the spin measurement record, introducing errors and limiting sensitivity. We show how to reduce this disturbance below δFα~ √N, the classical limit for N spins, by directing the QMBA almost entirely into an unmeasured spin component. This generates a planar squeezed state [1, 2] which allows simultaneous precise knowledge of spin angle and amplitude. We use high-dynamic-range optical quantum non-demolition measurements applied to a precessing magnetic spin ensemble, to demonstrate spin tracking with steady-state angular sensitivity 2.9 dB beyond the standard quantum limit, simultaneous with amplitude sensitivity 7.0 dB beyond Poisson statistics, surpassing classical limits in non-commuting observables for the first time [3]. Our method is close to practical application in high-performance atomic sensors, such as magnetometers and clocks, and is compatible with multi-pass and cavity build-up methods.
Summary form only given. Continuous monitoring or tracking of a quantum system is essential to high-sensitivity measurement of time-varying quantities. We estimate spin precession of a magnetically-sensitive atomic ensemble by measuring spin projectors Fαat different times. Such projectors do not commute, and thus quantum measurement back-action (QMBA) necessarily enters the spin measurement record, introducing errors and limiting sensitivity. We show how to reduce this disturbance below δFα~ √N, the classical limit for N spins, by directing the QMBA almost entirely into an unmeasured spin component. This generates a planar squeezed state [1, 2] which allows simultaneous precise knowledge of spin angle and amplitude. We use high-dynamic-range optical quantum non-demolition measurements applied to a precessing magnetic spin ensemble, to demonstrate spin tracking with steady-state angular sensitivity 2.9 dB beyond the standard quantum limit, simultaneous with amplitude sensitivity 7.0 dB beyond Poisson statistics, surpassing classical limits in non-commuting observables for the first time [3]. Our method is close to practical application in high-performance atomic sensors, such as magnetometers and clocks, and is compatible with multi-pass and cavity build-up methods.
Quantum entanglement, in the form of spin squeezing, is known to improve the sensitivity of atomic sensors to static or slowly varying fields. Sensing transient events presents a distinct challenge, requires different analysis tools, and has not been shown to benefit from entanglement in practically important scenarios such as spin-precession magnetometry. To address this, we apply concepts from continuous quantum measurements and estimation theory to optical atomic magnetometers, aiming to accurately model these devices, interpret their measurement data, control their dynamics, and achieve optimal sensitivity. Quantifying this optimal performance requires determining a fundamental quantum limit on sensitivity. We derive this limit, imposed by noise, and show that it scales at best linearly with sensing time and atom number N, ruling out any super-classical scaling. This limit is independent of the initial state, measurement, estimator, and measurement-based feedback, and depends only on the decoherence model and the strength of field fluctuations. Thus, finding an estimator that attains this bound proves the sensing strategy optimal. To approach this limit, we develop a quantum dynamical model scalable with N, based on a co-moving Gaussian approximation of the stochastic master equation, which includes measurement backaction and decoherence. This enables a real-time estimation and control architecture integrating an extended Kalman filter with a linear quadratic regulator. Simulating the magnetometer with our model and EKF+LQR strategy shows that quantum-limited tracking of constant and fluctuating fields is within reach of current atomic magnetometers. Our sensing strategy can also track biologically relevant signals, such as heartbeat-like waveforms, and drive the atomic ensemble into an entangled state, even when the measurement record is used for feedback but later discarded.
Measurements with quantum limited resolution have important applications in metrology and sensing, including atomic clocks, atom interferometry, and magnetometry. We perform a quantum-non-demolition measurement on a collective angular momentum of a million spin-4 Cs atoms with high optical depth in our work. Quantum mechanics limits the measurement's resolution to what is known as the standard quantum limit (SQL) when there is no entanglement between the individual atoms. We can generate entanglement between atoms to create spin squeezed states through quantum backaction, improving the measurement sensitivity by 5.5 dB relative to the SQL. We have recently rebuilt the entire experiment inside a state-of-the-art magnetic shield of mu-metal and aluminum. This upgrade has drastically mitigated classical control errors in the spin squeezing experiment. Furthermore, the ensemble's sensitivity to magnetic fields allows us to make precise measurements of the magnetic field environment. Using closed-loop feedback, we use our atomic magnetometer to measure magnetic fields 4.3 dB below the SQL. The shielded environment will allow us to optimize the magnetometer and sets the stage for future experiments to use the closed-loop control to explore more complicated quantum dynamics.
Quantum entanglement, in the form of spin squeezing, is known to improve the sensitivity of atomic instruments to static or slowly-varying quantities. Sensing transient events presents a distinct challenge, requires different analysis methods, and has not been shown to benefit from entanglement in practically-important scenarios such as spin-precession magnetometry (SPM). Here we adapt estimation control techniques introduced in [PRX Quantum 6, 030331 (2025)] to the experimental setting of SPM and analogous techniques. We demonstrate that real-time tracking of fluctuating fields benefits from measurement-induced spin squeezing and that quantum limits dictated by decoherence are within reach of today's experiments. We illustrate this quantum advantage by single-shot tracking, within the coherence time of a spin-precession magnetometer, of a magnetocardiography signal overlain with broadband noise.
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.
Large ensembles of uncorrelated atoms are extensively used as precise sensors of time, rotation, and gravity, and for tests of fundamental physics. The quantum nature of the sensors imposes a limit on their ultimate precision. Larger ensembles of N atoms can be used to average the quantum noise as 1/√N, a scaling known as the standard quantum limit. However, the ensemble size may be limited by technical constraints and/or atom-atom collisions -- a fundamental distinction from photon-based sensors. Learning to prepare entangled states of large ensembles with noise properties below the standard quantum limit will be key to extending both the precision and/or bandwidth of atomic sensors. More broadly, the generation and application of entanglement to solve problems is a core goal of quantum information science being pursued in both atomic and solid state systems.\nIn this thesis, we utilize the tools of cavity-QED to prepare entangled spin-squeezed states with 3.4(6) dB improvement in spectroscopic sensitivity over the standard quantum limit. The collective atomic spin is composed of the two-level clock states of 87Rb confined in a medium finesse F = 710 optical cavity. We employ cavity-aided quantum non-demolition measurements of the vacuum Rabi splitting to measure and subtract out the quantum projection noise of the collective spin state, preparing states with collective atomic spin projection noise 4.9(6) dB below the projection noise level. The conditionally reduced spin noise combined with the measured 1.5(3) dB reduction in the mean spin length implies a net 3.4(6) dB spectroscopic enhancement or conditional squeezing as defined by the Wineland criterion. Our method does not require single particle addressability and is applied to a spectroscopically large ensemble of N = 7 x 105 atoms using two collective population measurements, with the whole squeezing operation taking ~ 150 μs. The gain in sensitivity is spectroscopically equivalent to the enhancement obtained had we created > 105 pairs of maximally entangled qubits, demonstrating the power of a top-down approach for entangling large ensembles. The nondemolition probing of atomic populations via the vacuum Rabi splitting is also of broad interest for non-destructively reading out a wide variety of both atomic and solid state qubits.
This thesis describes quantum measurements of an ensemble of cold rubidium-87 atoms. We extend the covariance matrix formalism to spin-1 systems, including effects of decoherence, losses due to probing and atom number fluctuations. We show that the model can reproduce perimental results of both the mean and variance of a Faraday rotation measurement of the free induction decay signal of a coherent spin state precessing in an inhomogeneous magnetic field. We derive linearization procedures for Faraday measurements with high rotation angles and develop a fast differential photodetector that allows high dynamic range detection. We also study how to experimentally calibrate the reference quantum noise level under inhomogeneous light-atom interaction. We show that two non-commuting collective spin observables describing the atomic ensemble can be simultaneously known with sensitivity beyond classical limits, producing a planar quantum squeezed state. We theoretically study this state's metrological advantages and we find optimal conditions for its realization. Finally, using quantum non-demolition (QND) measurements of an atomic coherent state precessing under an orthogonal magnetic field, we track the radial and the angular component of the collective spin of the atomic ensemble below Poisson statistics and below the projection noise level by 7.0 dB and 2.9 dB respectively. The final topic of this thesis is the investigation of measurement backaction as a genuine quantum signature through the violation of Leggett-Garg inequalities. The use of QND measurements, which does not perturb the measured quantity, provide a way to certify such violations as due to true quantum effects rather than other possible classical disturvbance caused by the measurement. The use of Gaussian states described by covariance matrix calculations indicates these techniques can be applied to truly macroscopic systems. Esta tesis describe medidas cuánticas en una nube de átomos fríos de rubidio-87. Extendemos el formalismo de matriz de covariancia a sistemas de espin-1, incluyendo decoherencias, pérdidas debidas a la luz de prueba y fluctuaciones del número de átomos. Mostramos que el modelo puede reproducir los resultados experimentales de medias y variancias de medidas de rotación de Faraday de una señal de decaimiento libre de inducción de un estado coherente de spin rotando en un campo magnético. Derivamos procedimientos de linearizacion para medidas de Faraday con ángulos de rotación grandes y desarrollamos un fotodetector rápido que permite la detección en rango dinámico alto. En esta tesis estudiamos también el problema de interacción luz-átomos en el caso de acoplamiento inhomogéneo y calibramos experimentalmente el nivel de ruido cuántico. Demostramos que dos observables colectivas de espín que describen la nube atómica que no conmutan se pueden conocer simultáneamente con precisión más allá de los límites clásicos, produciendo un estado cuántico planar comprimido. Estudiamos teóricamente sus ventajas metrológicas y encontramos condiciones óptimas para su realización. Así mismo, utilizando medidas cuánticas no desctructivas (QND) en un estado atómico coherente rotando en un campo magnético ortogonal, monitorizamos las componentes radial y angular del spin colectivo de la nube de átomos 7.0 dB por debajo de la estadística Poissoniana y 2.9 dB por debajo del nivél del ruido de proyección. El último resultado de esta tesis es la investigación del contra-efecto de la medida como un auténtico efecto cuántico a través de la violación de las desigualdades de Leggett-Garg. El uso de las medidas QND, que no perturban la observable medida, permite considerar las violaciones como en verdadero efecto cuántico en lugar de otros posibles efectos clásicos causados por la medida. Por otro lado el uso de estados Gaussianos usado en los cálculos de matriz de covariancia puede ser aplicado también a sistemas verdaderamente macroscópicos.