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arXiv:1811.10088 · Physical Review A

Optimal estimation of matter-field coupling strength in the dipole approximation

József Zsolt Bernád, Claudio Sanavio, André Xuereb

Published Jun 12, 2019• 8 citations
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Abstract

This paper is devoted to the study of the Bayesian-inference approach in the context of estimating the dipole coupling strength in matter-field interactions. In particular, we consider the simplest model of a two-level system interacting with a single mode of the radiation field. Our estimation strategy is based on the emerging state of the two-level system, whereas we determine both the minimum mean-square error and maximum likelihood estimators for uniform and Gaussian prior probability density functions. In the case of the maximum likelihood estimator, we develop a mathematical method which extends the already existing approaches to the variational problem of the average cost function. We demonstrate that long interaction times, large initial mean photon numbers, and nonzero detuning between two-level system transition and the frequency of the electromagnetic field mode have a deleterious effect on the optimality of the estimation scenario. We also present several cases where the estimation process is inconclusive, despite many ideal conditions being met.

DOI
10.1103/physreva.99.062106

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We apply the formalism of quantum estimation theory to obtain information about the value of \nthe optomechanical coupling in the simplest model of two harmonic oscillators. In particular, we \ndiscuss the minimum mean-square error estimator and a quantum Cram er-Rao inequality for the \nestimation and accuracy of the coupling's value. Our estimation strategy reveals some cases, where \nquantum statistical inference is inconclusive and only prior expectations on the coupling strength \nare reassured. We show that this situations involve also the highest expected information losses. It \nis demonstrated that interaction times in the order of one time period of mechanical oscillations are \nthe most suitable for this type of estimation scenario. We also compare situations involving di erent \ninitial photon and phonon excitations.

•Dec 27, 2017•0 citations
This paper is devoted to the study of the Bayesian-inference approach in the context of estimating the dipole coupling strength in matter-field interactions. In particular, we consider the simplest model of a two-level system interacting with a single mode of the radiation field. Our estimation strategy is based on the emerging state of the two-level system, whereas we determine both the minimum mean-square error and maximum likelihood estimators for uniform and Gaussian prior probability density functions. In the case of the maximum likelihood estimator, we develop a mathematical method which extends the already existing approaches to the variational problem of the average cost function. We demonstrate that long interaction times, large initial mean photon numbers, and nonzero detuning between two-level system transition and the frequency of the electromagnetic field mode have a deleterious effect on the optimality of the estimation scenario. We also present several cases where the estimation process is inconclusive, despite many ideal conditions being met.
The formalism of quantum estimation theory with a specific focus on classical data postprocessing is applied to a two-level system driven by an external gyrating magnetic field. We employed both Bayesian and frequentist approaches to estimate the unknown transition frequency. In the frequentist approach, we have shown that only reducing the distance between the classical and the quantum Fisher information does not necessarily mean that the estimators as functions of the data deliver an estimate with desirable accuracy, as the classical Fisher information takes small values. We have proposed and investigated a cost function to account for the maximization of the classical Fisher information and the minimization of the aforementioned distance. Due to the nonlinearity of the probability mass function of the data on the transition frequency, the minimum variance unbiased estimator may not exist. The maximum likelihood and the maximum estimators often result in ambiguous estimates, which in certain cases can be made unambiguous upon changing the parameters of the external field. It is demonstrated that the minimum mean-square error estimator of the Bayesian statistics provides unambiguous estimates. In the Bayesian approach, we have also investigated the effects of noninformative and informative priors on the Bayesian estimates, including a uniform prior, Jeffrey's prior, and a Gaussian prior.
We consider a two-mode bosonic state with fixed photon number n∈Nn \in \mathbb {N}n∈N , whose upper and lower modes pick up a phase ϕ\phi ϕ and −ϕ-\phi −ϕ , respectively. We compute the optimal Fock coefficients of the input state, such that the mean square error (MSE) for estimating ϕ\phi ϕ is minimized, while the minimum MSE is always attainable by a measurement. Our setting is Bayesian, i.e., we consider ϕ\phi ϕ to be a random variable that follows a prior probability distribution function (PDF). Initially, we consider the flat prior PDF and we discuss the well-known fact that the MSE is not an informative tool for estimating a phase when the variance of the prior PDF is large. Therefore, we move on to study truncated versions of the flat prior in both single-shot and adaptive approaches. For our adaptive technique, we consider n=1n=1n=1 and truncated prior PDFs. Each subsequent step utilizes as prior PDF the posterior probability of the previous step, and at the same time we update the optimal state and optimal measurement.
We apply the formalism of quantum estimation theory to obtain information about the value of the nonlinear optomechanical coupling strength. In particular, we discuss the minimum mean-square error estimator and a quantum Cram\'er--Rao-type inequality for the estimation of the coupling strength. Our estimation strategy reveals some cases where quantum statistical inference is inconclusive and merely results in the reinforcement of prior expectations. We show that these situations also involve the highest expected information losses. We demonstrate that interaction times on the order of one time period of mechanical oscillations are the most suitable for our estimation scenario, and compare situations involving different photon and phonon excitations.
We establish, within the second quantization method, the general dipole-dipole Hamiltonian interaction of a system of nnn-level atoms. The variational energy surface of the nnn-level atoms interacting with $\ensuremath{\ell}$-mode fields and under the Van Der Waals forces is calculated with respect to the tensorial product of matter and electromagnetic field coherent states. This is used to determine the quantum phase diagram associated with the ground state of the system and quantify the effect of the dipole-dipole Hamiltonian interaction. By considering real induced electric dipole moments, we find the quantum phase transitions for 2- and 3-level atomic systems interacting with one and two modes of the electromagnetic field, respectively. The corresponding order of the transitions is established by means of Ehrenfest classification; for undetermined cases, we propose two procedures: one based on the difference of the expectation value of the Casimir operators of the 2-level subsystems, and another based on maximizing the Bures distance between neighbor variational solutions.
In conventional quantum optimal control theory, the parameters that determine an external field are optimised to maximise some predefined function of the trajectory, or of the final state, of a matter system. The situation changes in the case of quantum electrodynamics, where the degrees of freedom of the radiation field are now part of the system. In consequence, instead of optimising an external field, the optimal control question turns into a optimisation problem for the many-body initial state of the combined matter-photon system. In the present work, we develop such an optimal control theory for quantum electrodynamics. We derive the equation that provides the gradient of the target function, which is often the occupation of some given state or subspace, with respect to the control variables that define the initial state. We choose the well-known Dicke model to study the possibilities of this technique. In the weak coupling regime, we find that Dicke states are the optimal matter states to reach Fock number states of the cavity mode with large fidelity, and vice versa, that Fock number states of the photon modes are the optimal states to reach the Dicke states. This picture does not prevail in the strong coupling regime. We have also considered the extended case with more than one mode. In this case, we find that increasing the number of two-level systems allows reaching a larger occupation of entangled photon targets.
We apply the formalism of quantum estimation theory to obtain information about the value of \nthe optomechanical coupling in the simplest model of two harmonic oscillators. In particular, we \ndiscuss the minimum mean-square error estimator and a quantum Cram er-Rao inequality for the \nestimation and accuracy of the coupling's value. Our estimation strategy reveals some cases, where \nquantum statistical inference is inconclusive and only prior expectations on the coupling strength \nare reassured. We show that this situations involve also the highest expected information losses. It \nis demonstrated that interaction times in the order of one time period of mechanical oscillations are \nthe most suitable for this type of estimation scenario. We also compare situations involving di erent \ninitial photon and phonon excitations.