Squeezing of nonlinear spin observables by one axis twisting in the presence of decoherence: An analytical study

Abstract

In an ensemble of two-level atoms that can be described in terms of a collective spin, entangled states can be used to enhance the sensitivity of interferometric precision measurements. While non-Gaussian spin states can produce larger quantum enhancements than spin-squeezed Gaussian states, their use requires the measurement of observables that are nonlinear functions of the three components of the collective spin. In this paper we develop strategies that achieve the optimal quantum enhancements using non-Gaussian states produced by a nonlinear one-axis-twisting Hamiltonian, and show that measurement-afterinteraction techniques, known to amplify the output signals in quantum parameter estimation protocols, are effective in measuring nonlinear spin observables. Including the presence of the relevant decoherence processes from atomic experiments, we determine analytically the quantum enhancement of non-Gaussian over-squeezed states as a function of the noise parameters for arbitrary atom numbers.Résumé. Dans un ensemble d'atomes à deux niveaux descriptible par un spin collectif, on peut utiliser les états intriqués pour améliorer la sensibilité des mesures interférométriques. Bien que les états de spin non gaussiens puissent produire des améliorations quantiques plus importantes que les habituels état comprimés de spin, gaussiens, leur utilisation nécessite la mesure d'observables non linéaires en les trois composantes du spin collectif. Nous expliquons ici comment maximiser le gain quantique en utilisant des états non gaussiens surcomprimés produits par un hamiltonien non linéaire de torsion à un axe, et nous montrons que les techniques de mesure après intéraction, connues pour amplifier les signaux de sorties

Quantum-enhanced multiparameter estimation and compressed sensing of a field

“…For a time χ t L,best < χ t ≤ 1/ N , the quantum gain associated with (13), with an optimal choice of ⃗ n and ⃗ m, is larger than the gain associated with a linear measurement ξ −2 MAI > ξ −2 L,best [27]. It reaches its maximum value ξ −2 MAI,best at an optimal time χ t MAI,best = 1/ N in the large N limit [28].…”

“…This technique is equivalent to measuring a non-linear observable of the form XMAI = e −iχ t Ŝ2 z Ŝ ⃗ m e iχ t Ŝ2 z . It turns out that this measurement is optimal in the whole time range 1/N < χ t < 1/ N in the large N limit [26,27]. Also in this case, to first order in the θ ′ k in the vicinity of θ ′ k = 0, the average of the observable XMAI in the state Û′ |ψ t 〉 is…”

“…Equation (12) shows that the MAI technique allows the estimation of the linear combination Θ = k ε k θ k /N . In an estimation protocol based on the method of moments, the uncertainty on this combination is given by [27] (∆Θ)…”

“…The dependence of this parameter on N and χ t in the absence and presence of decoherence, with a linear measurement or by the MAI technique, is studied in detail in Ref. [27]. The question of the optimality of our strategy and a comparison with the quantum Fisher matrix in the framework of the theory of multiparameter estimation are presented in the Appendix B.…”

“…For the results in the presence of decoherence we used the analytical formula (9) and Appendix D of Ref. [27] giving the quantum gain obtained with a OAT state in the presence of a dephasing process (25). The scaling laws of the quantum gain as a function of N , in the absence and presence of decoherence, are given in the same reference [27].…”