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Photonics for Quantum (PfQ)

Optimal pulses for dipole moment estimation with coherent states

Karthik Chinni, Francesco Albarelli, Alexandre de Camargo, Rodrigo A. Vargas–Hernández, Nicolás Quesada

Published Jun 13, 2025• 0 citations
Publisher

Abstract

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 $omega{0}=delta pm frac{Gamma}{2}$ where $delta$ is the detuning and $Gamma$ is the spontaneous emission rate.

DOI
10.1117/12.3063316

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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=deltapmfracGamma2omega{0}=delta pm frac{Gamma}{2}omega0=deltapmfracGamma2 where deltadeltadelta is the detuning and GammaGammaGamma is the spontaneous emission rate.
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.
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.
Narrowband optimal control is proposed as a protocol of optimal quantum sensing, providing maximum accuracy for given resources. The procedure is demonstrated on the IBM quantum computer to estimate the amplitude of the qubit Rabi frequency. It is based on optimizing the quantum Fisher information (QFI) when it becomes equal to the classical Fisher information. This scenario offers a practical quantum sensing because it only involves population transfer to the excited state as the target without dynamical phases, which would otherwise be difficult to control and measure. The protocol uses the single-shot-shaped pulse method and an inverse optimization, to design a pulse that results in the optimal QFI in terms of energy or equivalently operating time, referred to in short as local narrowband inverse optimization (NIO). Multiple π pulses are shown to be particular cases of this protocol. An alternative semiglobal optimal NIO procedure producing a partially nonoscillatory profile is also demonstrated. Our approach also offers application to optimal atomic selective excitation in an array of atoms for quantum technology.
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.
This work discusses the problem of optimal excitation of a three-level atom of ladder configuration by light in the two-photon state and coherent light carrying an average of two photons. The applied atom-light interaction model is based on the Wigner-Weisskopf approximation. We characterize the properties of the optimal two-photon state that excites an atom perfectly, i.e., with probability equal to one. We find that the spectrotemporal shape of the optimal state of light is determined by the lifetimes of the atomic states, with the degree of photonic entanglement in the optimal state depending on the lifetime ratio. In consequence, two distinct interaction regimes can be identified in which the entanglement of the input state of light has a qualitatively different impact. As the optimal states may be challenging to prepare in general, we compare the results with those obtained for photon pairs of selected experimentally relevant pulse shapes. As these shapes are optimized for maximal atomic excitation probability, the results can be interpreted in terms of the overlap between the optimal and investigated pulse shapes.