Neste trabalho, apresentamos um amplo estudo das interações de longo alcance entre átomos de Rydberg frios na presença de campos elétricos estáticos. Nós observamos a transferência de população do estado quase molecular nD + nD para o estado (n+2)P após excitação pulsada na região de 29 ≤ n ≤ 41 em uma amostra de Rb aprisionada em uma armadilha magneto-óptica. A taxa de transferência pode ser manipulada com a presença de campo elétrico estático. Para explicar tais observações um modelo teórico multiníveis foi utilizado. O estudo de evolução temporal da população em (n+2)P mostrou que a dinâmica do processo é condizente com a interpretação clássica de uma transição diabática no domínio temporal. Utilizando um laser de excitação contínuo, realizamos experimentos envolvendo estados nD + nD, para 37 ≤ n ≤ 45, e estados nS + nS, para 39 ≤ n ≤ 47, onde foi possível estudarmos processos de transferência de população com resolução da estrutura fina. Também realizamos experimentos, tanto para estados nD quanto nS, para verificar a importância da estrutura hiperfina da estado fundamental no processo de transferência de população. / In this work, we present an extensive study of long-range interactions between cold Rydberg atoms in the presence of static electric fields. We have observed the population transfer from the quasi-molecular nD + nD state to the (n +2) P state after pulsed excitation for 29 ≤ n ≤ 41 in a sample of Rb trapped atoms in a magneto-optical trap. The transfer rate can be manipulated by the static electric field. To explain such observations, a multilevel theoretical model was used. The study of the time evolution of the population (n +2) P indicates that the dynamics of the process is consistent with a classical interpretation of a nonadiabatic transition in time domain. Using a CW laser excitation, experiments involving states nD + nD, for 37 ≤ n ≤ 45 states and nS + nS for 39 ≤ n ≤ 47, where performed with high spectral resolution. We also performed experiments for both states, nD and nS, to verify the importance of the hyperfine structure of the ground state in the process of population transfer.

Neste trabalho estudamos algumas interações entre átomos de Rydberg em uma armadilha ótica de dipolo do tipo QUEST. Com esta armadilha obtemos uma amostra de 1,2 × 106 átomos de 85Rb no estado fundamental, à uma densidade de ∼ 1012 átomos/cm3 e temperatura média de 60 µK. Os átomos de Rydberg foram preparados utilizando uma transição de dois fótons a partir do estado fundamental 5S1/2, passando pelo estado intermediário 5P3/2 e em seguida para o estado de Rydberg desejado. Estudamos a interação entre pares de átomos, em diferentes níveis energéticos, através de duas técnicas diferentes. Na primeira, monitoramos os efeitos de transferência de população em estados nD oriundas de uma ressonância Förster. Estudamos a ressonância nD5/2 + nD5/2 → (n+2)P3/2 + (n2)F7/2 onde 37 ≤ n ≤ 47, em função da densidade da amostra atômica; e para o estado 37D5/2 manipulamos a mesma ressonância com a aplicação de um campo elétrico externo. Os resultados mostraram que este é um processo binário, evidenciado pela dependência quadrática da população no estado produto com relação ao estado excitado. Num outro experimento, estudamos a interação entre estados nS através do monitoramento, e controle, do efeito de bloqueio de excitação. Aqui mostramos que é possível controlar as interações interatômicas em uma amostra quasi-unidimensional de átomos de Rydberg variando a orientação de um campo elétrico externo. Mostramos que ao polarizar uma amostra de átomos no estado 50S1/2 com um campo, esta passa a interagir de modo semelhante ao de dipolos elétricos clássicos, onde a interação pode ser controlada com a variação da orientação dos dipolos atômicos. Tal interação pode, inclusive, ser cancelada quando os dipolos elétricos são alinhados em um ângulo de 54,7° com relação ao eixo internuclear. / In this work we have studied some interactions between Rydberg-atoms in a QUEST type optical dipole trap. With this trap we obtained a sample of 1,2 × 106 85Rb atoms in the ground state, in a density of ∼ 1012 atoms/cm3 and average temperature of 60 µK. The Rydberg-atoms were prepared using a two-photon transition from the ground state 5S1/2, through an intermediate state 5P3/2 and then to the desired Rydberg state. We have studied interactions between pairs of atoms at several energy levels, using two different techniques. In the first one, we have monitored the effects of the population transfer in nD states derived from a Förster resonance. We have studied the resonance nD5/2 + nD5/2 → (n + 2)P3/2 + (n 2)F7/2 for states of 37 ≤ n ≤ 47 as a function of of the samples atomic density. For the 37D5/2 state we have also manipulated the same resonance with the application of an external electric field. Our results have shown that this is a binary process, indicated by the quadratic dependence of the transferred population in relation with the excited state. In another experiment, we have studied the interaction between nS1/2 states by monitoring, and controlling, the excitation blockade effect. Here we have shown that it is possible to control the inter-atomic interactions in a quasi-one-dimensional sample of Rydberg-atoms by varying the orientation of an external electric field. We have demonstrated that when polarizing a sample of atoms, in the 50S1/2 state with a field, it starts to interact in a similar way as classic electric-dipoles, where the interaction can be controlled by varying the orientation of the atomic dipoles. Such interaction may even be canceled when the electric dipoles are aligned at an angle of 54,7° related to the internuclear axis.

Neste trabalho estudamos a interação entre átomos de Rydberg no estado nP e sua dependência com o campo elétrico dc. Estes estados apresentam ressonâncias Föster para um número quântico principal n menor que 37. Nestes processos de ressonância um par atômico no estado nP muda para um par nS+(n+1)S. Realizamos dois experimentos de evolução temporal para 32≤n≤36. No primeiro investigamos a dependência da taxa de transferência de população NnS em função do número quântico principal n. E no segundo estudamos a transferência de população para um estado fixo de n=33 em função do campo elétrico. Além disso, estudamos a dependência da população no estado 33S em função da densidade de átomos de Rydberg no estado 33P. Estes resultados nos permitem observar duas contribuições distintas, uma linear relacionada a radiação de corpo negro e uma quadrática ligada a interação de dois corpos. Estes resultados confirmam o modelo de taxa para o efeito da radiação de corpo negro. / In this work we studied the role of Rydberg atoms interactions in the nP state and the dc electric field dependency of this process. The nP state shows Föster resonances for principal quantum number less than 37. In this resonance process, an atomic pair in nP state changes to a pair nS+(n+1)S. We have performed two time evolution experiments for 32≤n≤36. In the first one we have investigated the NnS population transfer rate for a variable principal quantum number n. In the second we have study the population transfer for a fixed n=33, by varying an electric field. Moreover, we observed the density dependency of the population in the 33S state by varying the nP state atomic density. The results allow us to observe two distinct contributions, a linear contribution related with the black body radiation and a quadratic one connected with two body process. The results agree well with the rate model used to treat the black body radiation.

In the atomic, molecular, optical physics field, processes involving two photons are very well understood and used in applications ranging from spectroscopy to laser cooling technics. In this thesis is presented the study and experimental realization of two two-photon processes. Using sodium atoms trapped in a magnetic optical trap we could demonstrate two-photon cooperative absorption, and that the creation of a molecular bound state using only light fields, or photonic bound state, is possible. Two-photon cooperative absorption, very common in solid-state physics, is a process where a pair of atoms initially in the ground state is excited to the double excited state, via absorption of two photons with frequecy that is not ressonant with any excited state. Its experimental realization with cold atoms may open new and exciting possibilities to better understand nonlinear effects, and it is a new way to create correlated atoms and photons in cold atomic physics. This absorption was observed by ionization of the pair after the excitation. A simple model that considers only dipole-dipole interactions between the atoms allows us to understand the basic features observed in the experimental data. A photonic bound state uses two photons to create the two basic features of a molecular bound state: a repulsive part and an attractive part. A blue photon, blue detuned from the atomic transition, connects the ground state of the pair to the repulsive part of the first excited molecular state 1, and a red photon, red detuned from the atomic transition, connects the connects the ground state of the pair to the attractive part of the first excited molecular state. In the dressed state picture, when the light fields are strong, this three-states-two-photon system creates adiabatic bound potentials that are strongly dependent of the photon properties. Using a theoretical model we could study how this bound energies changes when we change the photon properties, and the experimental data shows that this photos are indeed dressing the potentials with a efficiency that would enable the creation of photonic molecules. / No campo da física atômica, molecular e ótica processos envolvendo dois fótons são bem compreendidos e usados em diversas aplicações. Nesta tese apresentamos o estudo e a realização experimental de dois processos de dois-fótons. Usando átomos de sódio aprisionados em uma armadilha magneto ótica, demonstramos a absorção cooperativa de dois fótons e que a criação de um estado ligado molecular usando somente campos de luz, ou estado ligado fotônico, é possível. Absorção cooperativa de dois fótons, um processo bem comum em física de estado sólido, acontece quando um par de átomos inicialmente no estado fundamental é excitado para o estado duplamente excitado, via absorção de dois fótons de frequência não ressonante com a dos estados excitado. A realização experimental deste processo em um sistema de átomos frios pode abrir novas, e excitantes possibilidades para entender melhor processos não lineares, e é um novo método de criar átomos e fótons correlacionados. Essa absorção foi observada através da ionização do par depois da absorção, e um modelo simples que considera somente interação dipolo-dipolo entre os dois átomos nós ajuda entender as características básicas dos dados obtidos. Um estado ligado fotônico usa dois fótons para criar as duas características básicas dos estados ligados moleculares: a parte repulsiva e a parte atrativa. Um fóton azul, deslocado para o azul da transição atômica, conecta o estado fundamental do par a parte repulsiva do primeiro estado excitado molecular 1, e um fóton vermelho, deslocado para o vermelho da transição atômica, conecta o estado fundamental a parte atrativa do deslocado para o azul da transição atômica. No contexto de estados vestidos, quando os campos de luz são intensos, esse sistema de três estados e dois fótons cria potenciais ligantes adiabáticos que são fortemente dependentes das propriedades desses fótons. Usando um modelo teórico para esses potenciais pudemos estudar como é essa dependência, com as características do fótons, e os dados experimentais mostram que esses fótons estão de fato vestindo os estados com uma eficiência que viabiliza a criação de moléculas fotônicas.

Tendo por objetivo principal a implementação de um laboratório de pesquisa em tempo e freqüência, tomamos como primeiro alvo a ser alcançado a realização da definição primária do segundo, conforme estabelecido pelo BIPM, baseado no átomo de 133CS. Seguindo os passos nessa linha de pesquisa, o laboratório construiu um primeiro padrão baseado em feixe efusivo e operado opticamente. Atualmente, os sistemas com maior capacidade de determinação do segundo são os padrões de átomos frios, denominados chafarizes, dado o seu esquema de funcionamento. O objetivo principal desse trabalho foi o desenvolvimento de um padrão de tempo e freqüência baseado em um sistema de átomos frios, passando por todas as fases de sua implementação. Tais fases dizem respeito à construção de sistemas de controle de lasers de diodo, síntese de freqüências, avaliação de sinais de tempo e freqüência e controle e aquisição dos sinais de interrogação atômica. Adicionada a isso a necessidade de estabelecer um ambiente próprio ao desenvolvimento de tal experimento. Os resultados observados até aqui permitem que sejam estabelecidas novas metas, em busca do refinamento desse padrão. Além disso, os tipos de sistemas abordados são passíveis de utilização no desenvolvimento de outros padrões, tanto primários como secundários, sendo requisitos fundamentais no estabelecimento de metrologia científica de tempo e freqüência. / Having as a main goal the implementation of a research laboratory in time and frequency, we set the first target to be reached the realization of the primary definition of the second, as stated by BIPM, based on the 133CS atom. Following the steps in this research line our laboratory constructed a first standard based on an effusive beam optically operated. Nowadays, the most capable systems in the determination of the second are the cold atoms standards, called fountains, due to their operation mode. The main subject of this thesis is the development of a time and frequency standard based in cold atoms doing all the steps to implement it. These steps concern to the construction of diode lasers control systems, frequency synthesis, time and frequency signal characterization and atomic interrogation signal acquisition, added the need to the establishment of an appropriate environment to develop such experiment. The observed results until now allowed the determination of the new goals in the search of the standard refinement. Moreover, the kind of systems described here can be used in the development of other standards, either primary or secondary, being fundamental requisites in the establishment of time and frequency scientific metrology.

Cette thèse est consacrée aux progrès récents des horloges à réseau optique au strontium du LNE-SYRTE, Observatoire de Paris. L'incertitude systématique et la stabilité des horloges optiques sont 2 ordres de grandeur meilleures que les horloges atomiques micro-ondes au cesium qui réalisent la seconde SI, bénéficiant maintenent a des applications en physique fondamentale, astronomie et géosciences. Dans un futur proche, une redéfinition de la seconde SI est attendue, quand les horloges optiques se seront révélées aussi fiables et reproductibles que les horloges a micro-ondes. La thèse présente trois étapes décisives dans cette direction. Nous présentons un fonctionnemment operation quasi-continu de nos horloges Sr pendant plusieurs semaines. Des comparaisons de fréquences locales et à distance avec diverses références de fréquence micro-ondes et optiques montrent que les horloges optiques sont reproductibles par des laboratoires independants. Nous avons démontré un premier réseau tout optique entre des horloges optiques à l'échelle continentale. Les horloges au Sr ont été utilisées pour préparer 5 rapports de calibration du Temps Atomique International (TAI) qui ont été validés par le BIPM comme première contribution au TAI par des horloges optiques. Certains de ces résultats ont été utilisés pour borner l'amplitude d'une possible violation de l'invariance de Lorentz analysant les comparaisons d'horloges distantes. Enfin, nous avons effectué une caractérisation complète des déplacements de fréquence associés aux sources laser à semiconducteur utilisées pour le piégeage des atomes dans l'optique d'applications pour des horloges transportables et spatiales. / This thesis describes the latest progresses regarding the Sr optical lattice clocks at LNE-SYRTE, Observatoire de Paris. Nowadays, the systematic uncertainty and stability of optical clocks are 2 orders of magnitude better than cesium microwave fountains currently realizing the SI second, with applications in fundamental physics, astronomy and geoscience. In the near future, a re-definition of the SI second is expected, once optical clocks are proven to be as reliable and reproducible as their microwave counterparts. The thesis presents three decisive steps in this direction. First, we demonstrate nearly continuous Sr clocks over several weeks. Second, local and remote frequency comparisons against various microwave and optical frequency standards show that OLCs are reproducible over time, and by independent laboratories. We notably demonstrated the first all-optical agreement between optical clocks at continental scale. Third, the Sr clocks were used to calibrate the Temps Atomique International (TAI). The five calibration reports, which we produced, were validated by the BIPM, as the first contribution to TAI with optical clocks. In addition, some of these results were used to improve bounds on a putative violation of the Lorentz invariance by testing the stability of the frequency ratio between remote clocks. Finally, we conducted a full characterization of the frequency shifts associated with semi-conductor laser sources for the trapping light, including optical measurements and frequency shifts measurements, with applications for transportable and space clocks.

Mon travail de thèse se concentre sur l'étude, à l'aide de techniques numériques, de systèmes de fermions et bosons fortement corrélés. J'étudie Hamiltoniens de bosons sur réseau avec interactions à portée étendue, avant un intérêt pour expériences concernant atomes en états Rydberg-dressed, par moyen de simulations Path Integral Monte Carlo. Mon résultat principal est la démonstration d'un état de superverre en absence de sources de frustration dans le système.J'étudie également la modèle t-J fermionique avec deux trous par moyen de simulationsVariational Monte Carlo avec l’ansatz Entangled Plaquette States (EPS). Mon étude est fondamental en la perspective d'appliquer l'ansatz EPS à autres systèmes fermioniques, d’intérêt pour la supraconductivité à haute temperature, dont le comportement n'a pas encore été déterminé. Finalement, je présente mon travail sur une implémentation de l'algorithme Diagrammatic Monte Carlo. / The focus of my thesis is the investigation, via numerical approaches, of strongly correlated models of bosons and fermions. I study bosonic lattice Hamiltonians with extended--range interactions, of interest for experiments with cold Rydberg-dressed atoms, via Path Integral MonteCarlo simulations. My main result is the demonstration of a superglass in the absence of frustration sources in the system. I also study the fermionic $t-J$ model in the presence of two holes via Variational Monte Carlo with the Entangled Plaquette States Ansatz. My study is foundational to the extension of this approach to other fermionic systems, of interest for high temperature superconductivity, where the physical picture is still under debate (such as, e.g., the $t-J$ model in the case of finite hole concentration). Finally, I discuss my work on an implementation of the Diagrammatic Monte Carlo algorithm.

Dans cette thèse, nous faisons état de la construction d’un dispositif expérimentalcapable de piéger et refroidir un gaz d’ytterbium, dans l’optique de simuler des champsmagnétiques artificiels. Ce dispositif permettra, à terme, de produire et de caractériserdes états quantiques fortement corrélés, semblables aux états rencontrés dans la physiquede l’effet Hall quantique, entier ou fractionnaire.Dans un premier temps, nous décrivons la construction des parties de notre dispositifconsacrées au refroidissement optique de l’ytterbium (174Yb). En particulier, nousprésentons la conception d’un ralentisseur Zeeman, permettant le chargement direct d’unpiège magnéto-optique effectué sur la transition d’intercombinaison 1S0 ! 3P1 de l’ytterbium.Après transport dans un piège optique, une étape de refroidissement évaporatifnous permet d’obtenir des condensats de Bose-Einstein contenant environ 5×104 atomesd’ytterbium. Les condensats produits sont alors chargés dans un réseau optique verticalà la longueur d’onde « magique ».Nous présentons ensuite la construction d’un système laser étroit à 578nm capabled’exciter la transition « d’horloge » 1S0 ! 3P0 de l’ytterbium. Le laser est asservi surune cavité Fabry-Perot de grande finesse servant de référence de fréquence, dont nousavons caractérisé les différentes propriétés. Nous présentons en particulier une méthodepermettant de calibrer rapidement la fréquence absolue de la cavité par comparaison avecune transition de la molécule de diiode.Finalement, nous présentons les résultats d’expériences de spectroscopie effectuées surdes condensats d’ytterbium à l’aide du laser étroit, ainsi que la manipulation cohérentedu condensat sur la transition d’horloge au cours d’expériences d’oscillations de Rabi. Cesexpériences préliminaires ouvrent notamment la voie à la mesure des propriétés colisionnellesde l’ytterbium 174. / In this thesis, we report on the construction of an experiment aimed at trapping andcooling an ytterbium gaz, in order to realize artificial gauge fields. In the long term, thissetup will allow the study of strongly correlated quantum states which are atomic analogsof integer or fractional quantum Hall systems.We will first present the building of our experimental apparatus, and the optical coolingof ytterbium (174Yb). In particular, we will report on the design of a Zeeman slower,allowing for the direct loading of a magneto-optical trap operated on ytterbium’s intercombinationtransition 1S0 ! 3P1. The atomic cloud is then transported in an opticaldipole trap. A subsequent evaporative cooling stage results in the production of Bose-Einstein condensates of about 5 × 104 atoms.We then describe the construction of an ultra-narrow laser system at 578nm, able todrive ytterbium’s « clock » transition 1S0 ! 3P0. The laser frequency is stabilized using ahigh-finesse Fabry-Perot cavity, whose properties are precisely characterized in this work.Specifically, we present a method to calibrate the absolute frequency of the cavity bycomparison with an optical transition of molecular iodine.Finally, we show the results of spectroscopic measurements done on ytterbium condensatesusing the ultra-narrow laser. We also report on the coherent manipulation of thecondensate on the clock transition, consisting in the observation of Rabi oscillations.These preliminary experiments should allow for a measurement of ytterbium’s scatteringproperties.Keywords : cold atoms, optical lattices, Bose-Einstein condensates, ultra-stable lasers,clock transition, quantum simulation.

Au cours de ma thèse, j'ai travaillé à la construction de l'expérience Fermix, consacrée à l'étude d'un mélange de fermions (6Li-40K) à très basses températures où les effets quantiques sont prédominants. Nous présentons ici deux résultats principaux. Premièrement, nous avons développé une nouvelle méthode de refroidissement sub-Doppler qui tire parti de l'existence d'états noirs dans la raie optique D1. Cette mélasse grise permet d'atteindre une densité de l'espace des phases de 10^-4, la valeur la plus élevée rapportée dans la littérature pour le refroidissement laser simultané des deux espèces. Grâce à cette étape, nous avons pu réaliser un gaz fortement dégénéré de 3x10^5 atomes de 40K, répartis dans deux états de spins, à une température de 62 nK, soit 17% de la température de Fermi. D'autre part, nous introduisons une transformation canonique pour montrer l'équivalence formelle entre le comportement de particules ultra-relativistes sans interactions (particules de Weyl) dans un potentiel harmonique et celui de fermions froids confinés dans un piège quadrupolaire. Nous étudions expérimentalement, numériquement et théoriquement la relaxation de tels systèmes vers un état stationnaire, non-Botlzmanien, caractérisé par des températures effectives fortement anisotropes. Cette analogie permet également d'interpréter des propriétés caractéristiques des particules relativistes. Ainsi, nous montrons que le paradoxe de Klein est analogue aux pertes Majorana. Pour finir, nous proposons une étude théorique des interactions médiées à longue distance par un système en dimensions mixtes. / During my PhD, I contributed to the design and construction of the Fermix experiment, dedicated to the study of a 6Li-40K fermionic mixture at ultra low temperatures. Our main results are twofold. First, we developed a new sub-Doppler laser cooling scheme, taking advantage of the existence of dark states in the D1 line of alkali atoms. This so-called \emph{grey molasses} allows for a phase space density up to $10^{-4}$, the highest value reported for the simultaneous laser cooling of those two species. The improvement due to this cooling step enabled the production of a quantum degenerate 40K gas in a dipole trap, with 3x10^5 atoms in two spin states at 62 nK, corresponding to 17% of the Fermi temperature. Second, introducing a canonical mapping, we showed that non-interacting ultra-relativistic particles (Weyl fermions) in a harmonic trap can be simulated by cold fermions confined in a quadrupole potential. We study experimentally, numerically and theoretically the relaxation of these systems towards a steady state which can not be described by a Boltzman distribution, but rather presents strongly anisotropic effective temperatures. This analogy also allows us to translate fundamental properties of relativistic particles in the language of cold atoms. In particular, we demonstrate that the Klein paradox is equivalent to Majorana losses.Finally, we present a theoretical study of the long range interactions between particles confined in two 2D layers immersed in a 3D atomic cloud.

Intermédiaire essentiel au dialogue entre théorie et vérification expérimentale, la mesure n'a de sens que si la précision des résultats est élevée. La métrologie en laboratoire s'attache à augmenter autant que possible la précision avec laquelle l'expérimentateur a accès à la valeur d'un paramètre. Le bruit quantique affectant la mesure impose une limite sur la précision maximale accessible à partir d'états quasi-classiques: la limite quantique standard (SQL). La métrologie quantique cherche à utiliser les caractéristiques propres à la mécanique quantique pour la dépasser et se rapprocher le plus possible de la limite ultime, physiquement non franchissable, appelée limite de Heisenberg. Dans ce mémoire, nous avons développé une stratégie de mesure d'un champ électromagnétique contenant moins d'un photon basée sur l'utilisation de corrélations atome-champ dans une expérience d'électrodynamique quantique en cavité. L'idée est de mesurer l'amplitude de ce petit champ en sondant la perturbation qu'il introduit sur un état intriqué atome-champ mésoscopique déjà présent dans une cavité supraconductrice. Nous avons pu démontrer que le choix de notre mesure est, en principe, optimal grâce aux outils que sont l'information de Fisher (dépendant du processus de mesure) et l'information de Fisher dite quantique (qui elle n'en dépend pas), liées à la précision sur la mesure par des inégalités de type Cramér-Rao. Expérimentalement, nous avons très largement dépassé la précision obtenue sur l'amplitude du champ électromagnétique par une mesure classique et nous nous sommes rapprochés de la limite de Heisenberg autant que les imperfections expérimentales nous le permettaient. / As an essential intermediary between theories and their experimental proofs, measurement is meaningfull if the precision of its results is high. The main emphasis of metrology in laboratories is therefore on increasing as much as possible the precision of the experimental evaluation of a parameter. Quantum noise that affects the measurement establishes a quantitative limit on the maximal precision that can be achieved with classical states: the standard quantum limit (SQL). Quantum metrology aims at using quantum features to beat this limit and to approach the physically ultimate limit called Heisenberg limit. This thesis presents a measurement strategy for an electromagnetic field containing less than one photon, which is based on the use of atom-field correlations in a cavity quantum electrodynamics experiment. The idea is to measure the amplitude of the small field by probing the disturbance caused on an entangled mesoscopic state that is already stored in the superconducting cavity. We demonstrated that our measurement strategy is in principle optimal thanks to two tools: the Fisher information (that depends on the measurement process) and the quantum Fisher information (that does not), which define the precision tanks to Cramér-Rao like equations. The measurement signal subsequently largely exceeded the level of accuracy obtained with classical states and we got as closed to the Heisenberg limit as the experimental imperfections allowed us.