Continuous measurements play a fundamental role in quantum experiments, yet understanding and extracting the maximal possible information in a continuous probe signal remains a significant challenge. In this work, we show that for a wide class of sensors, the quantum Fisher information bound can be saturated through a combined measurement of signals from the original sensor and a suitable system copy. We elucidate the physical mechanism of such a twin-field sensor (TFS) and study its performance across diverse implementations, ranging from two-level atoms to a hybrid quantum Rabi model. In all of these cases, we find that simple photon counting or homodyne detection saturates the theoretical quantum limit. We further prove that the TFS achieves the quantum limits for the joint sensing of pairs of parameters as quantified by the quantum Fisher information matrix.
Continuous measurements play a fundamental role in quantum experiments, yet understanding and extracting the maximal possible information in a continuous probe signal remains a significant challenge. In this work, we show that for a wide class of sensors, the quantum Fisher information bound can be saturated through a combined measurement of signals from the original sensor and a suitable system copy. We elucidate the physical mechanism of such a twin-field sensor (TFS) and study its performance across diverse implementations, ranging from two-level atoms to a hybrid quantum Rabi model. In all of these cases, we find that simple photon counting or homodyne detection saturates the theoretical quantum limit. We further prove that the TFS achieves the quantum limits for the joint sensing of pairs of parameters as quantified by the quantum Fisher information matrix.
Precision measurements with quantum systems rely on our ability to trace the differences between experimental signals to variations in unknown physical parameters. In this Letter we derive the Fisher information and the ensuing Cramér-Rao sensitivity limit for parameter estimation by continuous measurements on an open quantum system. We illustrate our theory by its application to resonance fluorescence from a laser-driven two-state atom; we show that photon counting and homodyne detection records yield different sensitivity to the atomic parameters, while none of them exceed our general result.
Estimating a classical parameter encoded in the Hamiltonian of a quantum probe is a fundamental and well-understood task in quantum metrology. A textbook example is the estimation of a classical field’s amplitude using a two-level probe, as described by the semi-classical Rabi model. In this work, we explore the fully quantum analogue, where the amplitude of a coherent quantized field is estimated by letting it interact with a two-level atom. For both metrological scenarios, we focus on the quantum Fisher information (QFI) of the reduced state of the atomic probe. In the semi-classical Rabi model, the QFI is independent of the field amplitude and grows quadratically with the interaction time \tau τ . In contrast, when the atom interacts with a single coherent mode of the field, the QFI is bounded by 4, a constant dictated by the non-orthogonality of coherent states. We find that this bound can only be approached in the vacuum limit. In the limit of large amplitude \alpha α , the QFI is found to attain its maximal value 1.47 1.47 at \tau =O(1) τ = O ( 1 ) and \tau =O(\alpha^2) τ = O ( α 2 ) , and also shows periodic revivals at much later times. When the atom interacts with a sequence of coherent states, the QFI can increase with time but is bounded to scale linearly due to the production of entanglement between the atom and the radiation (back-action), except in the limit where the number of modes and their total energy diverge. Finally, in the continuous-field limit, where the atom interacts with a continuous source of weak coherent states, this back-action can be simply interpreted as spontaneous emission; we find that the optimal atomic QFI rate is finite, depends on the source intensity, and is upper bounded by the constant rate at which the QFI is emitted by the radiation source.
The quantum Cramer-Rao bound relates estimation sensitivity in quantum precision measurements to the distinguishability of different quantum states. We show that the theoretical sensitivity limit to parameters that govern the dynamics of an open quantum system coupled to a fully quantized environment can be evaluated from the reduced system master equation. We provide the bound for a laser driven two-state atom, where photon counting and homodyne detection of the fluorescence signal yield different sensitivity to the atomic and field parameters, while none of them exceed our general sensitivity limit.
The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of N resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with N better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources N is strictly constrained, I show that the final measurement uncertainty is π times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.
Abstract Quantum Fisher information quantifies the sensitivity encoded in a parameter-dependent quantum state, whereas the performance of a practical sensor is limited by the information that survives the physical channel and can be extracted by the available receiver. We introduce an accessible Fisher-information framework based on a physically admissible family of measurements and use it to separate state-level sensitivity, channel degradation, and receiver limitations. The framework is applied to Ramsey and NOON interferometry, squeezed-state displacement sensing, two-level thermometry, and multiparameter qubit sensing. We show that resource normalization, finite visibility, readout errors, measurement rank, shot allocation, calibration uncertainty, and nuisance parameters can substantially modify the experimentally available advantage. For squeezed-state sensing, homodyne-axis misalignment produces a finite optimal squeezing, while bounded calibration uncertainty leads to a robust operating rule. In the pure-loss model considered here, attenuation reduces the attainable information without shifting this optimum. The results provide practical design rules that connect quantum-enhanced state preparation with realistic measurement constraints and experimentally extractable sensing performance.
High-sensitivity accelerometers and gravimeters, achieving the ultimate limits of measurement sensitivity are key tools for advancing both fundamental and applied physics. While numerous platforms have been proposed to achieve this goal, from atom interferometers to optomechanical systems, all of these studies neglect the effects of intrinsic quantum uncertainty in time estimation. Starting from the Hamiltonian of a generic linear quantum sensor, we derive the two-parameter quantum Fisher information matrix and establish the corresponding Cram'er-Rao bound, treating time as an uncertain (nuisance) parameter. Our analysis reveals a fundamental coupling between time and signal estimation that inherently degrades measurement sensitivity, with the standard single-parameter quantum limit recovered only at specific interrogation times or under special decoupling conditions. We then apply these results to an optomechanical gravimeter and explicitly derive an optimal decoupling condition under which the effects of time uncertainty are averaged out in a continuous measurement scheme. Our approach is general and can be readily extended to a broad class of quantum sensors.