Abstract We investigate quantum sensing for spectroscopy in a system consisting of a two-level atom coupled to a continuum of modes. We focus on optimizing the pulse shape of a coherent state to maximize the quantum Fisher information (QFI) of the emitted light with the aim of estimating the atom’s dipole moment, which is proportional to its spontaneous emission rate. To achieve this, we derive a set of coupled differential equations, which include the standard optical Bloch equations as a subset and whose solution directly yields the QFI of the emitted light without resorting to finite-difference methods. Furthermore, we analyze the factors that govern its optimization, provide analytic solutions in both the long and the short pulse width limits, and examine the role of the average photon number of the pulses. We then show that under the closed (periodic) boundary conditions, the harmonic (plane-wave) with frequency equal to half the spontaneous emission rate and a phase determined by detuning are optimal in the long pulse width limit. We further show numerically that photodetection saturates the classical Fisher information.
Abstract We investigate quantum sensing for spectroscopy in a system consisting of a two-level atom coupled to a continuum of modes. We focus on optimizing the pulse shape of a coherent state to maximize the quantum Fisher information (QFI) of the emitted light with the aim of estimating the atom’s dipole moment, which is proportional to its spontaneous emission rate. To achieve this, we derive a set of coupled differential equations, which include the standard optical Bloch equations as a subset and whose solution directly yields the QFI of the emitted light without resorting to finite-difference methods. Furthermore, we analyze the factors that govern its optimization, provide analytic solutions in both the long and the short pulse width limits, and examine the role of the average photon number of the pulses. We then show that under the closed (periodic) boundary conditions, the harmonic (plane-wave) with frequency equal to half the spontaneous emission rate and a phase determined by detuning are optimal in the long pulse width limit. We further show numerically that photodetection saturates the classical Fisher information.
We investigate quantum sensing in spectroscopy by optimizing pulse shapes to maximize the quantum Fisher information (QFI) for estimating an atom’s dipole moment. Specifically, we provide the differential equations to compute the QFI of the light emitted from the interaction of coherent light with a two-level atom coupled to a waveguide and compute the optimal bandwidth for various families of standard pulses as a function of the average photon number. We provide intuition about the factors that maximize the QFI and show that the optimal pulse is a coherent state with frequency omega0=deltapmfracGamma2 where delta is the detuning and Gamma is the spontaneous emission rate.
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.
In recent years quantum metrology and quantum sensing have enabled the advancement of quantum technologies and research in fundamental physics. Through statistical analysis, in the context of parameter estimation, the amount of information about a parameter of interest encoded in a quantum state can be quantified. In this thesis we focus on the application of quantum estimation theory to imaging and spectroscopy with quantum light. We study the fundamental bound of the mean square error for an unbiased estimator in terms of the quantum Fisher information (QFI) for two problems, one in microscopy and one in spectroscopy. Firstly, we study the problem of localising multiple point sources below the diffraction limit. We show that localisation microscopy of multiple weak, incoherent point sources with possibly different intensities in one spatial dimension is equivalent to estimating the amplitudes of a classical mixture of coherent states of a simple harmonic oscillator. We obtain the QFI matrix elements analytically. In the regime of arbitrarily small separations we find it to be no more than rank two – implying that no more than two independent parameters can be estimated irrespective of the number of point sources. We use the eigenvalues of the classical and quantum Fisher information matrices to compare the performance of spatial-mode demultiplexing and direct imaging in localisation microscopy with respect to the quantum limits. Secondly, we study the estimation of the electric dipole moment (EDM) of a two-level atom through its interaction with quantum pulses of light in free space. We derive analytical expressions for the states of one-photon wavepackets and entangled photon pairs (EPP). We numerically calculate the QFI for different states of light, including coherent and squeezed pulses, and compare their performance in estimating the EDM. We find that the one-photon wavepacket and the EPP have similar performances, while the performance of the entangled photon pair is not improved by increasing the entanglement of the EPP. Our results indicate that using Fock states to estimate the EDM of a two-level atom is preferable to using entangled light.
We develop a quantum metrological framework for resonant nanophotonic sensors based on subwavelength Fabry-Perot (FP) slit cavities. Building directly on classical Fisher-information analyses of resonant transmission sensors, we model parameter encoding as a phase-and-loss quantum channel embedded in one arm of a Mach-Zehnder interferometer. We derive the quantum Fisher information (QFI) for coherent and Gaussian probe states under linear loss and show that, even at the quantum limit, optimal estimation precision is governed by the generator of parameter-dependent phase shifts-dominated by interface contributions-rather than by the cavity quality factor. Consequently, the operating point that maximizes QFI does not generally coincide with the maximum-Q resonance. Quantum resources enhance sensitivity but do not redefine the optimal geometry. Our results provide physically transparent design principles for quantum-enhanced nanophotonic sensing and clarify the central role of boundaries and loss in quantum-limited precision.
We find the optimal initial states of a linearized cavity-optomechanical system for estimating the single-photon coupling by quantum Fisher information. In the red-detuned (beam-splitter) regime the interaction exchanges excitations between the two modes, and for pure inputs the Fisher information is four times the variance of the excitation-exchange operator, reducing the optimization to variance maximization with a closed-form solution. With the mechanical mode in a Fock state the probe is degenerate: any state saturating the photon-number constraint is optimal, and a Fock reference with many phonons enhances the per-photon sensitivity linearly. For an arbitrary fixed mechanical reference it becomes a phase-matched squeezed vacuum, from an exact bound on that coherence obtained by a Lagrange-dual argument. Jointly optimizing both modes at fixed total excitation number, the quantum Fisher information obeys a Heisenberg bound quadratic in the energy, saturated by equal superpositions of the two extremal eigenstates of the generator, equivalently a two-mode NOON state after a balanced beam splitter; balanced product Fock states reach only a linear enhancement. Exact diagonalization in truncated Fock spaces confirms all the results, the familiar single-photon input being the simplest saturating case. Three extensions are worked out: the blue sideband has an unbounded two-mode-squeezing spectrum, so the entangled optimum becomes impossible and the balanced product Fock state is exactly optimal at definite energy; under cavity loss the information accumulation turns linear on the cavity-lifetime timescale, at a renewal-optimal duration of about two and a half lifetimes; and for a state-of-the-art quantum-coherent coupling experiment the per-hour relative precision lies between one part in a million and ten parts in a million, number and thermal references being the most robust.
Quantum mode parameter estimation determines parameters governing the shape of electromagnetic modes occupied by a quantum state of radiation. Canonical examples, time delays and frequency shifts, underpin radar, lidar, and optical clocks. A comprehensive framework recently established that broad families of quantum states can attain the Heisenberg limit, surpassing any classical strategy. This raises a fundamental question: among all quantum-enhanced strategies, which is truly optimal? Answering this requires identifying physically meaningful resources governing each estimation task, so quantum states can be compared on equal footing. We show these resources are connected to the eigenmode basis of the generator of the relevant mode transformation. For time-shift estimation, whose generator is diagonal in the frequency domain, the pertinent resources are the mean frequency and bandwidth; analogous quantities emerge for other transformations. Our framework unifies two historically separate perspectives: the particle-number aspect and the mode-structure of quantum light, providing a coherent picture of quantum-enhanced sensing with multimode radiation. Within this unified framework, we derive a tight upper bound on the quantum Fisher information for multimode Gaussian states, expressed in terms of these natural resources, and analytically identify the optimal Gaussian states saturating it. These optimal states take a particularly transparent form in the generator eigenbasis, a structural simplicity reflecting the deep connection between the geometry of the mode transformation and the architecture of the optimal probe. We further demonstrate that multimode homodyne detection constitutes the optimal measurement, achieving this bound and completing the end-to-end characterization of optimal quantum metrology strategies for mode parameter estimation.