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To extract as much information as possible about the state, we analyze various observables. We have already measured the second-order correlation functions; now we examine the 4-body correlation function and relative number squeezing. Additionally, we aim to confirm that the state is thermal so that we can apply the criterion derived in Chapter 2.","position":{"start":{"line":10,"column":1},"end":{"line":10,"column":1}},"key":"bHz3gYSYnC"}],"key":"n065izW7Q9"},{"type":"paragraph","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"children":[{"type":"text","value":"In the previous sections, we measured the value of the local correlation function and showed that it is compatible with 2. 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To further confirm this, we examine the single-mode statistics. In the first part of ","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"DKzlD8rd5W"},{"type":"crossReference","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"children":[{"type":"text","value":"subsection","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"kt9Xw8ZVaD"}],"identifier":"full_counting_stat","label":"full_counting_stat","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"full-counting-stat","key":"a4oaIajy8s"},{"type":"text","value":" ","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"S5VRrZewab"},{"type":"crossReference","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"children":[{"type":"text","value":"1","key":"wVoSjCmlgn"}],"identifier":"full_counting_stat","label":"full_counting_stat","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"full-counting-stat","key":"nI3Nat69p8"},{"type":"text","value":", we show a good agreement with thermal statistics by using additional datasets to vary the mean population of the state. We then compare the experimental distribution’s proximity to that of a slightly displaced state, and we compute the ","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"hJzXZ0lSBT"},{"type":"inlineMath","value":"N","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"html":"NNN","key":"B5q3B21unl"},{"type":"text","value":"-body local correlation function to highlight the dependence on thermal statistics.","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"vsvhaDqkJG"}],"key":"Posiq4PTYc"},{"type":"paragraph","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"In ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"CozvZsWqqL"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"subsection","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"V2l966J8Ac"}],"identifier":"pop_meas_sec","label":"pop_meas_sec","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"pop-meas-sec","key":"HtyOjvgxOv"},{"type":"text","value":" ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"F2aliBoShd"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"2","key":"FO1AsWFMMo"}],"identifier":"pop_meas_sec","label":"pop_meas_sec","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"pop-meas-sec","key":"FYG1jnDHwm"},{"type":"text","value":", we measure the population of the state using the mode size given by the width of the correlation functions. We can compare ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"AwVR7yOpqF"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"nJevarRnsx"},{"type":"text","value":" to the smaller bound that assesses entanglement, as discussed in ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"BlKXHrGebZ"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"the second chapter, section","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"YqSsitrObs"}],"identifier":"g2_fnc_sect","label":"g2_fnc_sect","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"g2-fnc-sect","remote":true,"url":"/entanglement-4correlation","dataUrl":"/entanglement-4correlation.json","key":"OqetVTQRxG"},{"type":"text","value":" ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"nvaIa4UjjC"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"2","key":"DJN2nGM0CV"}],"identifier":"g2_fnc_sect","label":"g2_fnc_sect","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"g2-fnc-sect","remote":true,"url":"/entanglement-4correlation","dataUrl":"/entanglement-4correlation.json","key":"b8s8DNs9X1"},{"type":"text","value":". We finally measure the 4-body correlation function and discuss how its value influence the non-separability of the state. In ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"x5THJ0h06G"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"subsection","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"tF7uMsd9P7"}],"identifier":"influence_efficiency","label":"influence_efficiency","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"influence-efficiency","key":"VuAOKZta0k"},{"type":"text","value":" ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"aTjFj6u4vP"},{"type":"crossReference","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"4","key":"eyYQiGysBo"}],"identifier":"influence_efficiency","label":"influence_efficiency","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"influence-efficiency","key":"ANy4Qjc4i4"},{"type":"text","value":", we discuss the influence of the non-unit quantum efficiency of the detector. In this section, we also measure the normalized variance and compare it to the expected value.","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"Lp4jT7RTft"}],"key":"nr0YgprjPP"},{"type":"paragraph","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"children":[{"type":"text","value":"Finally, we conclude in ","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"key":"ctOJgkWFb7"},{"type":"crossReference","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"children":[{"type":"text","value":"subsection","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"key":"HsU9KQrtaW"}],"identifier":"conclu_ici","label":"conclu_ici","kind":"heading","template":"Section %s","enumerator":"6","resolved":true,"html_id":"conclu-ici","key":"motzImBp7g"},{"type":"text","value":" ","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"key":"cBRpuk3HSL"},{"type":"crossReference","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"children":[{"type":"text","value":"6","key":"ejipBh26xR"}],"identifier":"conclu_ici","label":"conclu_ici","kind":"heading","template":"Section %s","enumerator":"6","resolved":true,"html_id":"conclu-ici","key":"raHEd0tIip"},{"type":"text","value":" on the presence of entanglement. We also report measurements of the Cauchy-Schwarz ratio across various excitation durations. Initially, entanglement is detected, but at later times, it is no longer observable.","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"key":"azGilGI8KD"}],"key":"ViFPRp1CkT"},{"type":"comment","value":"In the previous sections, we measured the value of the local correlation function and showed that they are compatible with 2. In the discussion of chapter 2, [section](#ccl_entanglement) [%s](#ccl_entanglement), we underlined that the criterion derived is correct only if the state is not displaced and that $\\braket{\\hat{a}_k^2}=0$. To further demonstrate this is the case, we look at the single mode statistics. In the first part of [subsection](#full_counting_stat) [%s](#full_counting_stat), we show a good agreement of thermal statistics using other datasets to change the mean population of the state. We ten compare the distance of the experimental distribution to that of a slightly displaced state, and we compute the $N$-body local correlation function to evidence the thermal statistics dependence.","position":{"start":{"line":22,"column":1},"end":{"line":22,"column":1}},"key":"thpVhYWHOm"}],"data":{"part":"abstract"},"key":"C4JAEt2HwK"},{"type":"block","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"children":[{"type":"heading","depth":2,"position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"children":[{"type":"text","value":"Full counting statistics of a single mode","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"mMxDTb55tl"}],"identifier":"full_counting_stat","label":"full_counting_stat","html_id":"full-counting-stat","enumerator":"1","key":"qopXPpTcb5"},{"type":"heading","depth":3,"position":{"start":{"line":28,"column":1},"end":{"line":28,"column":1}},"children":[{"type":"text","value":"Fock probability distribution","position":{"start":{"line":28,"column":1},"end":{"line":28,"column":1}},"key":"b8LdXBkpJ9"}],"identifier":"fock-probability-distribution","label":"Fock probability distribution","html_id":"fock-probability-distribution","implicit":true,"key":"JiNN2qb8pR"},{"type":"paragraph","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"children":[{"type":"text","value":"We begin by demonstrating our ability to fully resolve the statistical properties of an individual mode. Here, we analyze a different dataset in which the excitation duration was varied. ","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"adxEswRRAY"},{"type":"crossReference","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"children":[{"type":"text","value":"Figure ","key":"Ik9G3uhGjQ"},{"type":"text","value":"1","key":"hCE74oMKJB"}],"identifier":"full_counting_satatistics","label":"full_counting_satatistics","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"full-counting-satatistics","key":"StWDjbXaxd"},{"type":"text","value":" reports such probability distribution for four different mean populations. Our goal is to emphasize that, whatever the population, the single mode statistic is always well-described by a thermal statistics.","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"G8tw6sVhGv"}],"key":"hpu9emT4i1"},{"type":"comment","value":"In the last subsection, we integrated over a large volume, and we lose information about the mode momentum, *e.g*. its \"position\" in $k$-space. Now, we select a particular mode to investigate its atom number distribution. However, it is necessary to define the \"size\" of this mode. Drawing an analogy from optics, we need to filter out unwanted modes using pinholes. In our experiment, this filtering is achieved via post-selection, which defines the size of the 3D box within which we compute the statistics. Just as \"pixel\" refers to a unit cell in 2D images, here we use the term *voxel* to refer to a 3D volume (in $k$-space). The $k$-position of this voxel is set at the density maximum of the peaks. The theoretical size of this voxel is $2\\pi/\\vec{L}$, where $\\vec{L}$ represents the BEC length. While this quantity is not known [precisely](#bec_size_measure), we can use the correlation length measured in the [last subsection](#local_correlation_function).","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"key":"OYtLO4lsMc"},{"type":"container","kind":"figure","identifier":"full_counting_satatistics","label":"full_counting_satatistics","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/full_counting_satati-143cca6b3822bef44fe325341c3cb5e1.png","alt":"Comparison of the the density profile and second order correlation function along the y direction","width":"100%","align":"center","key":"qfkVsonYpV","urlSource":"images/full_counting_satatistics.png","urlOptimized":"/~gondret/phd_manuscript/build/full_counting_satati-143cca6b3822bef44fe325341c3cb5e1.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"full_counting_satatistics","identifier":"full_counting_satatistics","html_id":"full-counting-satatistics","enumerator":"1","children":[{"type":"text","value":"Figure ","key":"jo0NGo2YAb"},{"type":"text","value":"1","key":"f9v9tLliTW"},{"type":"text","value":":","key":"hTN0nFED0c"}],"template":"Figure %s:","key":"tW1mZGGUUr"},{"type":"text","value":"Full counting statistics of the sidebands. We plot the probability distribution function of the atom number. The yellow circles show the experimental statistics. The solid red line and dashed blue line show the thermal and Poissonian distributions, calculated based on the measured mean atom number with no free parameter. 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Each panel shows the probability distribution for a different excitation duration, hence a different mean atom number. Regardless of the population, the probability distribution remarkably agrees with a thermal distribution over three orders of magnitude. 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Note that the fraction ","position":{"start":{"line":51,"column":1},"end":{"line":51,"column":1}},"key":"KkA3tDr0Hn"},{"type":"inlineMath","value":"f_{th}","position":{"start":{"line":51,"column":1},"end":{"line":51,"column":1}},"html":"fthf_{th}fth​","key":"thPhJSfbqa"},{"type":"text","value":" is insensitive to the detection efficiency","position":{"start":{"line":51,"column":1},"end":{"line":51,"column":1}},"key":"AibjGI6ZBR"},{"type":"footnoteReference","identifier":"effet_detect","label":"effet_detect","position":{"start":{"line":51,"column":1},"end":{"line":51,"column":1}},"number":3,"enumerator":"3","key":"pYryUctWOX"},{"type":"text","value":" ","position":{"start":{"line":51,"column":1},"end":{"line":51,"column":1}},"key":"JkO6TGcDCZ"},{"type":"inlineMath","value":"\\eta ","position":{"start":{"line":51,"column":1},"end":{"line":51,"column":1}},"html":"η\\eta η","key":"uyacKg0JGZ"},{"type":"text","value":", as it only affects the mean population ","position":{"start":{"line":51,"column":1},"end":{"line":51,"column":1}},"key":"YuXePMZDkb"},{"type":"inlineMath","value":"\\bar{n}_{det}=\\eta\\bar{n}","position":{"start":{"line":51,"column":1},"end":{"line":51,"column":1}},"html":"nˉdet=ηnˉ\\bar{n}_{det}=\\eta\\bar{n}nˉdet​=ηnˉ","key":"vYVNA0PpyD"},{"type":"text","value":" and not ","position":{"start":{"line":51,"column":1},"end":{"line":51,"column":1}},"key":"TRI34rElvO"},{"type":"inlineMath","value":"f_{th}","position":{"start":{"line":51,"column":1},"end":{"line":51,"column":1}},"html":"fthf_{th}fth​","key":"rWGLfDr4lR"},{"type":"text","value":". This ensures that the analysis directly probes the state’s relative displacement.","position":{"start":{"line":51,"column":1},"end":{"line":51,"column":1}},"key":"wGRCaIvLqw"}],"key":"VBCvY68NLz"},{"type":"paragraph","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"children":[{"type":"text","value":"Panel (a) of ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"XGdjjdrrj1"},{"type":"crossReference","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"children":[{"type":"text","value":"Figure ","key":"fHGD3qqD7N"},{"type":"text","value":"2","key":"toxHDKN5MG"}],"identifier":"g_norder_fthermal","label":"g_norder_fthermal","kind":"figure","template":"Figure %s","enumerator":"2","resolved":true,"html_id":"g-norder-fthermal","key":"PXA8ayIvQL"},{"type":"text","value":" shows the distance between the two distributions as a function of the thermal fraction ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"LWCpVOpEmp"},{"type":"inlineMath","value":"f_{th}","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"fthf_{th}fth​","key":"e4vT1t4YmJ"},{"type":"text","value":". The positive mode is represented in orange and the negative mode in blue. For the triangles, the longitudinal size of the voxel is larger than the mode size (2.7 mm/s). For the square and round markers, the voxel size is 0.7 mm/s, which is smaller than a mode size.","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"ddwu6kNfq7"}],"key":"R5BRe2C6gZ"},{"type":"paragraph","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"children":[{"type":"text","value":"We observe the maximum distance consistently occurs at ","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"key":"QyXm0gtwPT"},{"type":"inlineMath","value":"f_{th}=0","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"html":"fth=0f_{th}=0fth​=0","key":"IcIeYdEzKS"},{"type":"text","value":". It means that the probability distribution is never well described by a fully coherent state, which was already clear from ","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"key":"Ai7nSZN2G2"},{"type":"crossReference","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"children":[{"type":"text","value":"Figure ","key":"DBttI1ASha"},{"type":"text","value":"1","key":"Yd5UAzcBhW"}],"identifier":"full_counting_satatistics","label":"full_counting_satatistics","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"full-counting-satatistics","key":"xyKvXqre4J"},{"type":"text","value":". However, we observe that the minimum for each curve occurs at different thermal fraction ","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"key":"A6KfseJbSA"},{"type":"inlineMath","value":"f_{th}^{min}","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"html":"fthminf_{th}^{min}fthmin​","key":"iT1BkfOjtS"},{"type":"text","value":". Notably, for larger voxel sizes, the distribution is neither thermal nor Poissonian and ","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"key":"NeYGYL6iLm"},{"type":"inlineMath","value":"f_{th}^{min} \\sim 0.5","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"html":"fthmin∼0.5f_{th}^{min} \\sim 0.5fthmin​∼0.5","key":"QTRdqtz5Em"},{"type":"text","value":". For such a voxel size, we might in fact counting several (thermal) modes. 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When the mean population per mode ","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"key":"ECvIW4rwvU"},{"type":"inlineMath","value":"\\bar{n}","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"html":"nˉ\\bar{n}nˉ","key":"vwvfhsVUYX"},{"type":"text","value":" is the same, the probability distribution is known and is a function of ","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"key":"rYMgTzid5r"},{"type":"inlineMath","value":"\\bar{n}","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"html":"nˉ\\bar{n}nˉ","key":"yEBfKxtEQr"},{"type":"text","value":" and the number of modes ","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"key":"IKfU15XRC6"},{"type":"inlineMath","value":"m","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"html":"mmm","key":"dkXhsdyG5m"},{"type":"text","value":". 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The solid line represents ","position":{"start":{"line":75,"column":1},"end":{"line":75,"column":1}},"key":"AjfSIQOEYy"},{"type":"inlineMath","value":"n!","position":{"start":{"line":75,"column":1},"end":{"line":75,"column":1}},"html":"n!n!n!","key":"UFyJDl2kuH"},{"type":"text","value":", which is the expected value for a thermal state. Blue triangles and red circles respectively show the correlation value for the negative and positive peaks. For the negative peak, a significant discrepancy is observed for ","position":{"start":{"line":75,"column":1},"end":{"line":75,"column":1}},"key":"Iy7uWrRd3T"},{"type":"inlineMath","value":"n>4","position":{"start":{"line":75,"column":1},"end":{"line":75,"column":1}},"html":"n>4n>4n>4","key":"bBFeqswo16"},{"type":"text","value":". On the other hand, for the positive peak, we observe a remarkable (and surprising) agreement. Indeed, we calculate the correlation function in a single voxel with a 0.7 mean population. To determine ","position":{"start":{"line":75,"column":1},"end":{"line":75,"column":1}},"key":"fTjnteU2PR"},{"type":"inlineMath","value":"g^{(4)}","position":{"start":{"line":75,"column":1},"end":{"line":75,"column":1}},"html":"g(4)g^{(4)}g(4)","key":"H2rAlM2kmR"},{"type":"text","value":", we analyze coincidence counts that involve at least 4 atoms simultaneously. For a thermal distribution and 1400 realization, the expected number of cycles with at least 4 detected atoms is 40. It decreases to 7 cycles for ","position":{"start":{"line":75,"column":1},"end":{"line":75,"column":1}},"key":"Bz9dZDuQK3"},{"type":"inlineMath","value":"g^{(6)}","position":{"start":{"line":75,"column":1},"end":{"line":75,"column":1}},"html":"g(6)g^{(6)}g(6)","key":"MHukezQXAz"},{"type":"text","value":" and only 3 cycles for ","position":{"start":{"line":75,"column":1},"end":{"line":75,"column":1}},"key":"o9M84THdPM"},{"type":"inlineMath","value":"g^{(7)}","position":{"start":{"line":75,"column":1},"end":{"line":75,"column":1}},"html":"g(7)g^{(7)}g(7)","key":"VwZIYrqFtY"},{"type":"text","value":".","position":{"start":{"line":75,"column":1},"end":{"line":75,"column":1}},"key":"OQtgJFmf7r"}],"key":"B7ayzgVGcQ"},{"type":"paragraph","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"children":[{"type":"text","value":"To assess entanglement, the criterion derived in the second chapter’s ","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"key":"MejWyDmUnq"},{"type":"crossReference","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"children":[{"type":"text","value":"section","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"key":"MnVQsQueL5"}],"identifier":"g2_fnc_sect","label":"g2_fnc_sect","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"g2-fnc-sect","remote":true,"url":"/entanglement-4correlation","dataUrl":"/entanglement-4correlation.json","key":"dIr1Vc9NP8"},{"type":"text","value":" ","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"key":"JGJDd9WXRy"},{"type":"crossReference","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"children":[{"type":"text","value":"2","key":"vcq9MebhI3"}],"identifier":"g2_fnc_sect","label":"g2_fnc_sect","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"g2-fnc-sect","remote":true,"url":"/entanglement-4correlation","dataUrl":"/entanglement-4correlation.json","key":"teB5ZW8cKh"},{"type":"text","value":" requires that the un-displaced Gaussian state also to satisfy ","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"key":"DlpEhcZrl0"},{"type":"inlineMath","value":"\\braket{\\hat{a}_k^2}=0","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"html":"⟨a^k2⟩=0\\braket{\\hat{a}_k^2}=0⟨a^k2​⟩=0","key":"AxEEcdbJqz"},{"type":"text","value":". Using Wick expansion and introducing","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"key":"C7PrVdLWrC"},{"type":"footnoteReference","identifier":"confusion","label":"confusion","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"number":4,"enumerator":"4","key":"l4JfFO8bmS"},{"type":"text","value":" ","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"key":"YEfA86xrtx"},{"type":"inlineMath","value":"\\alpha_k := \\braket{\\hat{a}^2_k}/\\braket{\\hat{a}_k^\\dagger\\hat{a}_k}","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"html":"αk:=⟨a^k2⟩/⟨a^k†a^k⟩\\alpha_k := \\braket{\\hat{a}^2_k}/\\braket{\\hat{a}_k^\\dagger\\hat{a}_k}αk​:=⟨a^k2​⟩/⟨a^k†​a^k​⟩","key":"VTdTmEDWki"},{"type":"text","value":", one can show that","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"key":"I9JdyJOU0l"}],"key":"Gpj7TvgN6g"},{"type":"math","identifier":"influence_alpha","label":"influence_alpha","value":"g^{(2)}_{k,k} = 2 +|\\alpha|^2, \\quad \\quad g^{(3)}_{k,k} = 6 + 9|\\alpha|^2, \\quad \\quad g^{(4)}_{k,k} = 24 + 72|\\alpha|^2 + 9|\\alpha|^4.","tight":true,"html":"gk,k(2)=2+∣α∣2,gk,k(3)=6+9∣α∣2,gk,k(4)=24+72∣α∣2+9∣α∣4.g^{(2)}_{k,k} = 2 +|\\alpha|^2, \\quad \\quad g^{(3)}_{k,k} = 6 + 9|\\alpha|^2, \\quad \\quad g^{(4)}_{k,k} = 24 + 72|\\alpha|^2 + 9|\\alpha|^4.gk,k(2)​=2+∣α∣2,gk,k(3)​=6+9∣α∣2,gk,k(4)​=24+72∣α∣2+9∣α∣4.","enumerator":"2","html_id":"influence-alpha","key":"CP6u1Rj0Yh"},{"type":"paragraph","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"children":[{"type":"text","value":"Based on the error bars","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"key":"rmuD1PsUm9"},{"type":"footnoteReference","identifier":"bootstrap_g2g3g4","label":"bootstrap_g2g3g4","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"number":5,"enumerator":"5","key":"fHb7HRLIKo"},{"type":"text","value":" in ","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"key":"mGqQmTNx3z"},{"type":"crossReference","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"children":[{"type":"text","value":"Figure ","key":"lbNMwygNWM"},{"type":"text","value":"2","key":"zoioGK8Wbc"}],"identifier":"g_norder_fthermal","label":"g_norder_fthermal","kind":"figure","template":"Figure %s","enumerator":"2","resolved":true,"html_id":"g-norder-fthermal","key":"dLRLqt7Oci"},{"type":"text","value":"(b), the measurement of ","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"key":"S0fzlkpvgN"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"j7ky395ttZ"},{"type":"text","value":" is consistent with 2.0 with an uncertainty of 0.1. 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Using these uncertainties, we estimate that ","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"key":"u7Qa4dige8"},{"type":"inlineMath","value":"|\\alpha|^2<0.1","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"html":"∣α∣2<0.1|\\alpha|^2<0.1∣α∣2<0.1","key":"yklQk5xJbc"},{"type":"text","value":" using ","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"key":"bITTlBLRFK"},{"type":"inlineMath","value":"g^{(2)}_{k,k}","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"html":"gk,k(2)g^{(2)}_{k,k}gk,k(2)​","key":"E3WE1n8hER"},{"type":"text","value":". 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Here, all these measurements agree that ","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"key":"uQ8mhl1z1r"},{"type":"inlineMath","value":"|\\alpha|^2<0.1","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"html":"∣α∣2<0.1|\\alpha|^2<0.1∣α∣2<0.1","key":"BTUxDoQJDL"},{"type":"text","value":". In the previous sections, we measured the local correlation with two methods, finding its value consistent with 2.00 with an uncertainty of 0.05. 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The mean value is therefore 2.24(7). 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We reintroduce the notations of the ","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"U3ZwRw9RMv"},{"type":"crossReference","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"children":[{"type":"text","value":"second chapter, section","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"l0qIIk9uhy"}],"identifier":"what_info_cov_matrix","label":"what_info_cov_matrix","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"what-info-cov-matrix","remote":true,"url":"/entanglement-4correlation","dataUrl":"/entanglement-4correlation.json","key":"nN0d6fcpvH"},{"type":"text","value":" ","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"EbnpeDGgRs"},{"type":"crossReference","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"children":[{"type":"text","value":"1","key":"ct9vNBL6lz"}],"identifier":"what_info_cov_matrix","label":"what_info_cov_matrix","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"what-info-cov-matrix","remote":true,"url":"/entanglement-4correlation","dataUrl":"/entanglement-4correlation.json","key":"qtkjrsl1s3"},{"type":"text","value":" and define","position":{"start":{"line":162,"column":1},"end":{"line":162,"column":1}},"key":"wTB5DU4vYl"}],"key":"uJKkcpeCs9"},{"type":"math","identifier":"definition_c_dbis","label":"definition_c_dbis","value":"c_k := \\braket{\\hat{a}_{k}\\hat{a}_{-k}}, \\quad \\quad \\quad \nd_k := \\braket{\\hat{a}_{k}\\hat{a}_{-k}^\\dagger}.","tight":true,"html":"ck:=⟨a^ka^−k⟩,dk:=⟨a^ka^−k†⟩.c_k := \\braket{\\hat{a}_{k}\\hat{a}_{-k}}, \\quad \\quad \\quad \nd_k := \\braket{\\hat{a}_{k}\\hat{a}_{-k}^\\dagger}.ck​:=⟨a^k​a^−k​⟩,dk​:=⟨a^k​a^−k†​⟩.","enumerator":"7","html_id":"definition-c-dbis","key":"RyJqbRc39z"},{"type":"paragraph","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"children":[{"type":"text","value":"When ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"mAwozxUbCe"},{"type":"inlineMath","value":"g^{(4)}","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"g(4)g^{(4)}g(4)","key":"RjpR39C6Pp"},{"type":"text","value":" is close to its minimal value in Eq. 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The fact that the second order correlation function is above 2 and that the population is high makes impossible the second scenario: such state would not respect the ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"a5LM4IHWRi"},{"type":"emphasis","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"sH47f7TZk0"}],"key":"R0U8B9vwGG"},{"type":"text","value":" condition. We have therefore ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"Gdrt9ov7wk"},{"type":"inlineMath","value":"|c_k|>|d_k|","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"∣ck∣>∣dk∣|c_k|>|d_k|∣ck​∣>∣dk​∣","key":"Y1rXRxbQDP"},{"type":"text","value":". 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In the ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"HBskQc2MQ8"},{"type":"crossReference","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"children":[{"type":"text","value":"second chapter","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"PombMyEDXI"}],"identifier":"what_info_cov_matrix","label":"what_info_cov_matrix","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"what-info-cov-matrix","remote":true,"url":"/entanglement-4correlation","dataUrl":"/entanglement-4correlation.json","key":"V15EllZi91"},{"type":"text","value":", we also defined ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"FokPku2o2G"},{"type":"inlineMath","value":"\\theta ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"θ\\theta θ","key":"HCnOMLfbEX"},{"type":"text","value":" in Eq. 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We show in ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"LvR6IeGBz8"},{"type":"crossReference","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"children":[{"type":"text","value":"Figure ","key":"HcliDZD0UK"},{"type":"text","value":"5","key":"B6YO2cGTsT"}],"identifier":"g4donnees_experimentale","label":"g4donnees_experimentale","kind":"figure","template":"Figure %s","enumerator":"5","resolved":true,"html_id":"g4donnees-experimentale","key":"QRK0oH89Bw"},{"type":"text","value":" the value of the logarithmic negativity as a function of ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"SAvZUZP13o"},{"type":"inlineMath","value":"\\theta ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"θ\\theta θ","key":"D0IEhnqAbX"},{"type":"text","value":". The solid purple line uses ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"uUzhwhxj7x"},{"type":"inlineMath","value":"g^{(2)}_{k,-k} = 2.24","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"gk,−k(2)=2.24g^{(2)}_{k,-k} = 2.24gk,−k(2)​=2.24","key":"alpv3OYG96"},{"type":"text","value":" and the dotted curves its lower and higher uncertainty. The line stops at ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"Pf4kFY61sW"},{"type":"inlineMath","value":"\\theta\\sim 0.3","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"θ∼0.3\\theta\\sim 0.3θ∼0.3","key":"fjO8S2vppY"},{"type":"text","value":" due to the ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"IO0fOSGqvn"},{"type":"emphasis","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"children":[{"type":"text","value":"bona fide","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"ySyi5bzotX"}],"key":"W5Qg69fi05"},{"type":"text","value":" condition: for such high value of ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"mFlxuuVDdT"},{"type":"inlineMath","value":"g^{(2)}_{k,-k}","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"gk,−k(2)g^{(2)}_{k,-k}gk,−k(2)​","key":"PeOCLAaHGn"},{"type":"text","value":" and the measured population, it is not possible for the value of ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"jdTGBPMOpO"},{"type":"inlineMath","value":"|d_k|","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"∣dk∣|d_k|∣dk​∣","key":"YxrJgZlRIK"},{"type":"text","value":" to be too high. The position of this region depends on the value of ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"xOrvtrWSfG"},{"type":"inlineMath","value":"g^{(2)}_{k,-k}","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"gk,−k(2)g^{(2)}_{k,-k}gk,−k(2)​","key":"iFWpqQywk1"},{"type":"text","value":": the dotted curve for which ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"brOVae2e23"},{"type":"inlineMath","value":"g^{(2)}=2.17","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"g(2)=2.17g^{(2)}=2.17g(2)=2.17","key":"dYn5JUjtLN"},{"type":"text","value":" has a critical value for ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"c5FerWroUh"},{"type":"inlineMath","value":"\\theta ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"θ\\theta θ","key":"T97ns79N96"},{"type":"text","value":" of ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"rYNEQ37PnW"},{"type":"inlineMath","value":"\\sim 0.4","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"∼0.4\\sim 0.4∼0.4","key":"dJDLwiGKog"},{"type":"text","value":" while it is 0.25 for ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"XujGqmoxYl"},{"type":"inlineMath","value":"g^{(2)}_{k,-k}=2.31","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"gk,−k(2)=2.31g^{(2)}_{k,-k}=2.31gk,−k(2)​=2.31","key":"J6jbjuebRT"},{"type":"text","value":". Our measurement of ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"niWm3693G1"},{"type":"inlineMath","value":"g^{(4)}_{+-}","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"g+−(4)g^{(4)}_{+-}g+−(4)​","key":"kZDg65BkyH"},{"type":"text","value":" is more compatible with ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"yRUGVSWvOd"},{"type":"inlineMath","value":"\\theta=0","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"θ=0\\theta=0θ=0","key":"gmmAgBe1WT"},{"type":"text","value":". It is therefore more likely that the logarithmic negativity of the state is 0.4(1). We see however from this figure that a non-zero value of ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"PFKFB5eaxK"},{"type":"inlineMath","value":"d_k","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"html":"dkd_kdk​","key":"F35XVMQfnu"},{"type":"text","value":" cannot ","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"ocaGHkgkr3"},{"type":"emphasis","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"children":[{"type":"text","value":"de-entangle","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"ukxbQnCXIb"}],"key":"edatH4STCC"},{"type":"text","value":" our state. In fact, it can only increase entanglement as seen here.","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"AM1LOTtkkN"}],"key":"cgeNTFZRWB"},{"type":"container","kind":"figure","identifier":"g4donnees_experimentale","label":"g4donnees_experimentale","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/g4donnees_experiment-dbbe04a9c06a0a0c67c3fa9f2cc7374a.png","alt":"LogNeg","width":"50%","align":"center","key":"J2uX1vMjYk","urlSource":"images/g4donnees_experimentale.png","urlOptimized":"/~gondret/phd_manuscript/build/g4donnees_experiment-dbbe04a9c06a0a0c67c3fa9f2cc7374a.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":177,"column":1},"end":{"line":177,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"g4donnees_experimentale","identifier":"g4donnees_experimentale","html_id":"g4donnees-experimentale","enumerator":"5","children":[{"type":"text","value":"Figure ","key":"lWgLFAO02j"},{"type":"text","value":"5","key":"u98sasJnj2"},{"type":"text","value":":","key":"MXtQQsVM4s"}],"template":"Figure %s:","key":"CHt0jIPK39"},{"type":"text","value":"Logarithmic negativity of the state for which ","position":{"start":{"line":177,"column":1},"end":{"line":177,"column":1}},"key":"Yc5pkFFmK4"},{"type":"inlineMath","value":"g^{(2)}=2.24(7)","position":{"start":{"line":177,"column":1},"end":{"line":177,"column":1}},"html":"g(2)=2.24(7)g^{(2)}=2.24(7)g(2)=2.24(7)","key":"p4IP241Hww"},{"type":"text","value":" as a function of ","position":{"start":{"line":177,"column":1},"end":{"line":177,"column":1}},"key":"CcxPBZbSM4"},{"type":"text","value":"θ","position":{"start":{"line":177,"column":1},"end":{"line":177,"column":1}},"key":"LT9QbagRwJ"},{"type":"text","value":". 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In the next section, we take into account the efficiency of the detector to reconstruct the state before the detector.","position":{"start":{"line":180,"column":1},"end":{"line":180,"column":1}},"key":"drBvoeqi8n"}],"key":"TexRaaMlKv"},{"type":"comment","value":"Taking into account this efficiency makes impossible the further that $\\theta = 0$.\nNote that this is what we expect theoretically: $\\braket{\\hat{a}_{k}\\hat{a}_{-k}^\\dagger}=0$. From this value, we obtain the logarithmic negativity of the state: 0.4(1).","position":{"start":{"line":181,"column":1},"end":{"line":182,"column":1}},"key":"Ufmrfe1dpK"},{"type":"heading","depth":2,"position":{"start":{"line":187,"column":1},"end":{"line":187,"column":1}},"children":[{"type":"text","value":"Influence of the non-unit efficiency","position":{"start":{"line":187,"column":1},"end":{"line":187,"column":1}},"key":"S6VkGxyXvK"}],"identifier":"influence_efficiency","label":"influence_efficiency","html_id":"influence-efficiency","enumerator":"4","key":"OYM2Wv2ppS"},{"type":"paragraph","position":{"start":{"line":188,"column":1},"end":{"line":188,"column":1}},"children":[{"type":"text","value":"We now assume that ","position":{"start":{"line":188,"column":1},"end":{"line":188,"column":1}},"key":"S7Jk5pjKKx"},{"type":"inlineMath","value":"\\braket{\\hat{a}_{k}\\hat{a}_{-k}^\\dagger}=0","position":{"start":{"line":188,"column":1},"end":{"line":188,"column":1}},"html":"⟨a^ka^−k†⟩=0\\braket{\\hat{a}_{k}\\hat{a}_{-k}^\\dagger}=0⟨a^k​a^−k†​⟩=0","key":"aNm13pVHfx"},{"type":"text","value":", which is a reasonable assumption given our measurement of ","position":{"start":{"line":188,"column":1},"end":{"line":188,"column":1}},"key":"TlG1GClOWE"},{"type":"inlineMath","value":"g^{(4)}_{k,-k}","position":{"start":{"line":188,"column":1},"end":{"line":188,"column":1}},"html":"gk,−k(4)g^{(4)}_{k,-k}gk,−k(4)​","key":"VMCRCZnmWh"},{"type":"text","value":". Additionally, we assume the detected mean populations are identical and equal to 0.94. We now study the influence of the non-unit detection efficiency.","position":{"start":{"line":188,"column":1},"end":{"line":188,"column":1}},"key":"HaItNXEHVz"}],"key":"bhpfI2REzs"},{"type":"heading","depth":3,"position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"children":[{"type":"text","value":"Parametrization of the state","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"key":"nVM6e6ILBA"}],"identifier":"parametrization-of-the-state","label":"Parametrization of the state","html_id":"parametrization-of-the-state","implicit":true,"key":"TvgnXxGenr"},{"type":"paragraph","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"children":[{"type":"text","value":"Based on our simplifications, the state is simply parametrized by two numbers : ","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"key":"Hts9um6rKI"},{"type":"inlineMath","value":"c_k","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"html":"ckc_kck​","key":"u18h5PxdJg"},{"type":"text","value":" and ","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"key":"XbUAw4uUYA"},{"type":"inlineMath","value":"n_k","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"html":"nkn_knk​","key":"Wm3GheeSgg"},{"type":"text","value":". In the following, to lighten notations, I assume that ","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"key":"WZXcpGq37B"},{"type":"inlineMath","value":"c_k","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"html":"ckc_kck​","key":"W18wVtA4ZR"},{"type":"text","value":" is real and positive (it involves otherwise the modulus of ","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"key":"X6CksuqGH8"},{"type":"inlineMath","value":"c_k","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"html":"ckc_kck​","key":"Tz2VkNzAfP"},{"type":"text","value":"). The second order correlation function that we measured is not affected by losses and provides us the value of ","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"key":"GqAPfvB2f6"},{"type":"inlineMath","value":"c_k^{(det)}","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"html":"ck(det)c_k^{(det)}ck(det)​","key":"db3QfYHoZG"},{"type":"text","value":" through the value of ","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"key":"lBsgCedHeH"},{"type":"inlineMath","value":"n_k^{(det)}=0.94","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"html":"nk(det)=0.94n_k^{(det)}=0.94nk(det)​=0.94","key":"vZbxP2RWUO"},{"type":"text","value":":","position":{"start":{"line":190,"column":1},"end":{"line":190,"column":1}},"key":"augsA66xmU"}],"key":"xbG31iGBIq"},{"type":"math","value":"c_k = n_k\\sqrt{g^{(2)}-1}","tight":true,"html":"ck=nkg(2)−1c_k = n_k\\sqrt{g^{(2)}-1}ck​=nk​g(2)−1​","enumerator":"8","key":"c5xL2tlPwP"},{"type":"paragraph","position":{"start":{"line":194,"column":1},"end":{"line":194,"column":1}},"children":[{"type":"text","value":"Effect of losses can be modelled by mixing each mode with the vacuum on a beam-splitter with efficiency ","position":{"start":{"line":194,"column":1},"end":{"line":194,"column":1}},"key":"fzhiWMoUTy"},{"type":"inlineMath","value":"\\eta ","position":{"start":{"line":194,"column":1},"end":{"line":194,"column":1}},"html":"η\\eta η","key":"xDeMOHxi8f"},{"type":"text","value":". As seen in the ","position":{"start":{"line":194,"column":1},"end":{"line":194,"column":1}},"key":"R1o1eFmJqR"},{"type":"crossReference","position":{"start":{"line":194,"column":1},"end":{"line":194,"column":1}},"children":[{"type":"text","value":"second chapter, section","position":{"start":{"line":194,"column":1},"end":{"line":194,"column":1}},"key":"fitf9bs3Hr"}],"identifier":"single_mode_transfo","label":"single_mode_transfo","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"single-mode-transfo","remote":true,"url":"/entanglement-1gaussian","dataUrl":"/entanglement-1gaussian.json","key":"bTeTluy6se"},{"type":"text","value":" ","position":{"start":{"line":194,"column":1},"end":{"line":194,"column":1}},"key":"CSt33n6don"},{"type":"crossReference","position":{"start":{"line":194,"column":1},"end":{"line":194,"column":1}},"children":[{"type":"text","value":"4","key":"Oo7N64DWPl"}],"identifier":"single_mode_transfo","label":"single_mode_transfo","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"single-mode-transfo","remote":true,"url":"/entanglement-1gaussian","dataUrl":"/entanglement-1gaussian.json","key":"wxnXfQa0mX"},{"type":"text","value":" it changes the covariance matrix ","position":{"start":{"line":194,"column":1},"end":{"line":194,"column":1}},"key":"EZSZwLT6r3"},{"type":"inlineMath","value":"\\boldsymbol{\\sigma} \\rightarrow \\eta\\boldsymbol{\\sigma} + (1-\\eta) \\mathbb{I}_2","position":{"start":{"line":194,"column":1},"end":{"line":194,"column":1}},"html":"σ→ησ+(1−η)I2\\boldsymbol{\\sigma} \\rightarrow \\eta\\boldsymbol{\\sigma} + (1-\\eta) \\mathbb{I}_2σ→ησ+(1−η)I2​","key":"HJIpjdL6Az"},{"type":"text","value":". With our parametrization, effect of losses is even simpler:","position":{"start":{"line":194,"column":1},"end":{"line":194,"column":1}},"key":"VdwuXdBtF5"}],"key":"CcO0bywBzh"},{"type":"math","value":"n_k = n_k^{(det)}/\\eta\n\\quad ,\\quad \\quad c_k = c_k^{(det)}/\\eta.","tight":true,"html":"nk=nk(det)/η,ck=ck(det)/η.n_k = n_k^{(det)}/\\eta\n\\quad ,\\quad \\quad c_k = c_k^{(det)}/\\eta.nk​=nk(det)​/η,ck​=ck(det)​/η.","enumerator":"9","key":"iWnmT6zgzk"},{"type":"paragraph","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"children":[{"type":"text","value":"The state we describe is a two-mode squeezed thermal state. Physically, it is better to describe it with two other numbers: the squeezing parameter ","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"key":"BEiOQfjCtg"},{"type":"inlineMath","value":"r_k","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"html":"rkr_krk​","key":"ZOGzMev10U"},{"type":"text","value":" and the initial (thermal) mode occupancy ","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"key":"e3AE6NL4Mr"},{"type":"inlineMath","value":"n_{k,th}^{(in)}","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"html":"nk,th(in)n_{k,th}^{(in)}nk,th(in)​","key":"dsc67A05vE"},{"type":"text","value":". The covariance matrix of this state was also discussed in ","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"key":"gwuMrClFJc"},{"type":"crossReference","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"children":[{"type":"text","value":"chapter 2, section","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"key":"FwOmix2Vn5"}],"identifier":"section_bi_partite_gaussian_state_transofrmations","label":"section_bi_partite_gaussian_state_transofrmations","kind":"heading","template":"Section %s","enumerator":"5","resolved":true,"html_id":"section-bi-partite-gaussian-state-transofrmations","remote":true,"url":"/entanglement-1gaussian","dataUrl":"/entanglement-1gaussian.json","key":"P7Eadrea38"},{"type":"text","value":" ","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"key":"QUSbHEncOB"},{"type":"crossReference","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"children":[{"type":"text","value":"5","key":"P8SF4Hdv2Y"}],"identifier":"section_bi_partite_gaussian_state_transofrmations","label":"section_bi_partite_gaussian_state_transofrmations","kind":"heading","template":"Section %s","enumerator":"5","resolved":true,"html_id":"section-bi-partite-gaussian-state-transofrmations","remote":true,"url":"/entanglement-1gaussian","dataUrl":"/entanglement-1gaussian.json","key":"OPtLWVdZER"},{"type":"text","value":". Nonetheless, we can access the value of ","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"key":"T1ynPd8UBd"},{"type":"inlineMath","value":"n_{k,th}^{(in)}","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"html":"nk,th(in)n_{k,th}^{(in)}nk,th(in)​","key":"J3qDT2Zgge"},{"type":"text","value":" and ","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"key":"oUOBZWWLTB"},{"type":"inlineMath","value":"r_k","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"html":"rkr_krk​","key":"uouyap4LJ5"},{"type":"text","value":" through the value of ","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"key":"HCPMCgHENN"},{"type":"inlineMath","value":"c_k","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"html":"ckc_kck​","key":"HcDhG8d1Fh"},{"type":"text","value":" and ","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"key":"eSJNv1gjtx"},{"type":"inlineMath","value":"n_k","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"html":"nkn_knk​","key":"uqsmDlP52Z"},{"type":"text","value":":","position":{"start":{"line":199,"column":1},"end":{"line":199,"column":1}},"key":"vkL6dHwXuw"}],"key":"X9cTif9Ek3"},{"type":"math","identifier":"squeezing_param_initial_temp","label":"squeezing_param_initial_temp","value":"\\text{tanh}(2r_k) = \\frac{2c_k}{2n_k+1}\\quad , \\quad \\quad 2n_{k,th}^{(in)}+1=\\frac{2n_k+1}{\\text{cosh}(2r_k)}.","tight":true,"html":"tanh(2rk)=2ck2nk+1,2nk,th(in)+1=2nk+1cosh(2rk).\\text{tanh}(2r_k) = \\frac{2c_k}{2n_k+1}\\quad , \\quad \\quad 2n_{k,th}^{(in)}+1=\\frac{2n_k+1}{\\text{cosh}(2r_k)}.tanh(2rk​)=2nk​+12ck​​,2nk,th(in)​+1=cosh(2rk​)2nk​+1​.","enumerator":"10","html_id":"squeezing-param-initial-temp","key":"mlKLHuzVqv"},{"type":"paragraph","position":{"start":{"line":204,"column":1},"end":{"line":204,"column":1}},"children":[{"type":"text","value":"and the value of the logarithmic negativity is given by","position":{"start":{"line":204,"column":1},"end":{"line":204,"column":1}},"key":"rk9npiDFFm"}],"key":"TgNGEy0Vyg"},{"type":"math","identifier":"lognegdefentang","label":"lognegdefentang","value":"E_\\mathcal{N} = -\\log_2\\left[(2n_{k,th}^{(in)}+1)e^{-2r_k}\\right].","tight":true,"html":"EN=−log⁡2[(2nk,th(in)+1)e−2rk].E_\\mathcal{N} = -\\log_2\\left[(2n_{k,th}^{(in)}+1)e^{-2r_k}\\right].EN​=−log2​[(2nk,th(in)​+1)e−2rk​].","enumerator":"11","html_id":"lognegdefentang","key":"jURs0xQEm2"},{"type":"paragraph","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"children":[{"type":"text","value":"In ","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"key":"ApRtGLH7cp"},{"type":"crossReference","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"children":[{"type":"text","value":"Figure ","key":"neTrGsERZ5"},{"type":"text","value":"6","key":"RqOAxJl1i7"}],"identifier":"resume_entanglement","label":"resume_entanglement","kind":"figure","template":"Figure %s","enumerator":"6","resolved":true,"html_id":"resume-entanglement","key":"n7ZgTMUS6h"},{"type":"text","value":", we plot some state properties as a function of the quantum efficiency of the detector ","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"key":"wG7Jfy0nOG"},{"type":"inlineMath","value":"\\eta ","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"html":"η\\eta η","key":"w6b0FXgBc0"},{"type":"text","value":", which range between 20 and 50%. Each curve represents a different value of the second order correlation function, taken in the uncertainty range of ","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"key":"wLnLPYq2lB"},{"type":"inlineMath","value":"g^{(2)}=2.24(7)","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"html":"g(2)=2.24(7)g^{(2)}=2.24(7)g(2)=2.24(7)","key":"ZGVlon9Odt"},{"type":"text","value":".","position":{"start":{"line":209,"column":1},"end":{"line":209,"column":1}},"key":"wFuvWLSTiS"}],"key":"vAg1woFncX"},{"type":"paragraph","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"children":[{"type":"text","value":"Panel (a) shows the state mean population: the smaller the efficiency, the larger the mean occupation of the reconstructed state. The markers show the position where each curve stops for the corresponding value of ","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"key":"wN0s54qw8q"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"Jzn60avTfI"},{"type":"text","value":": the highest the value of ","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"key":"a4oOYTKaGC"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"k2eevp4NVr"},{"type":"text","value":", the smallest the value ","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"key":"KqHfQYyAUK"},{"type":"inlineMath","value":"\\eta_{min}","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"html":"ηmin\\eta_{min}ηmin​","key":"vkO7GQdpiI"},{"type":"text","value":" for which our parametrization is physical. Indeed, for a given value of ","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"key":"STliHNp4Vn"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"TWxlt1huEG"},{"type":"text","value":", the Gaussian state with the highest mean population that has this value of ","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"key":"YLBPUGvtwR"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"LinijDdif1"},{"type":"text","value":" is the two-mode squeezed vacuum state. For this state, we have ","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"key":"zwsLSS6CDQ"},{"type":"inlineMath","value":"g^{(2)}=2+1/\\bar{n}","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"html":"g(2)=2+1/nˉg^{(2)}=2+1/\\bar{n}g(2)=2+1/nˉ","key":"wtEzlENfWw"},{"type":"text","value":". The minimal value of the efficiency is therefore ","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"key":"FZmrLQX0AH"},{"type":"inlineMath","value":"\\eta_{min}=(g^{(2)}_{k,-k}-2)n_{k}^{(det)}","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"html":"ηmin=(gk,−k(2)−2)nk(det)\\eta_{min}=(g^{(2)}_{k,-k}-2)n_{k}^{(det)}ηmin​=(gk,−k(2)​−2)nk(det)​","key":"JEiL7pD3yj"},{"type":"text","value":". Hence, the larger ","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"key":"FRId7NlTSf"},{"type":"inlineMath","value":"g^{(2)}_{k,-k}","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"html":"gk,−k(2)g^{(2)}_{k,-k}gk,−k(2)​","key":"Ezbt9NQrUQ"},{"type":"text","value":", the larger the minimal efficiency. With ","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"key":"ciGB6BGh1K"},{"type":"inlineMath","value":"g^{(2)}=2.24","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"html":"g(2)=2.24g^{(2)}=2.24g(2)=2.24","key":"UXShNufnUj"},{"type":"text","value":", we see that the quantum efficiency is at least 23%, and if we consider the lower uncertainty, it is at least 17%.","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"key":"DHZaBjJvfX"}],"key":"NHDunFckMD"},{"type":"paragraph","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"children":[{"type":"text","value":"Panels (b) and (d) of ","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"key":"mvpu09Xdzl"},{"type":"crossReference","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"children":[{"type":"text","value":"Figure ","key":"zc2kG7Uvk3"},{"type":"text","value":"6","key":"J1JTYdtflo"}],"identifier":"resume_entanglement","label":"resume_entanglement","kind":"figure","template":"Figure %s","enumerator":"6","resolved":true,"html_id":"resume-entanglement","key":"uasJfBrzOs"},{"type":"text","value":" show respectively the logarithmic negativity and the squeezing parameter ","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"key":"KRn4ARHx8j"},{"type":"inlineMath","value":"r_k","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"html":"rkr_krk​","key":"WI9fK3K0TM"},{"type":"text","value":". We observe that they monotonically increase with ","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"key":"roRwkw9Vgj"},{"type":"inlineMath","value":"1/\\eta","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"html":"1/η1/\\eta1/η","key":"P07PaRpUJc"},{"type":"text","value":". Here again, the smaller the quantum efficiency, the larger the ","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"key":"BSvAQQpFwk"},{"type":"emphasis","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"children":[{"type":"text","value":"real","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"key":"TTjQWpBKBU"}],"key":"sWj13ESlrS"},{"type":"text","value":" state and the larger its correlations.","position":{"start":{"line":214,"column":1},"end":{"line":214,"column":1}},"key":"oTULDGdhTs"}],"key":"WdNuOZ4CHM"},{"type":"container","kind":"figure","identifier":"resume_entanglement","label":"resume_entanglement","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/resume_entanglement-4784edb79b7f77111e79fb300f8a84a5.png","alt":"LogNeg","width":"100%","align":"center","key":"OPdKkEZ0pm","urlSource":"images/resume_entanglement.png","urlOptimized":"/~gondret/phd_manuscript/build/resume_entanglement-4784edb79b7f77111e79fb300f8a84a5.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":222,"column":1},"end":{"line":222,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"resume_entanglement","identifier":"resume_entanglement","html_id":"resume-entanglement","enumerator":"6","children":[{"type":"text","value":"Figure ","key":"XluhI7z11U"},{"type":"text","value":"6","key":"SEQ2AhiK8E"},{"type":"text","value":":","key":"gr2HOnTKwm"}],"template":"Figure %s:","key":"rjCZoIYJqo"},{"type":"text","value":"(a) Mean population of the state taking into account the detection efficiency ","position":{"start":{"line":222,"column":1},"end":{"line":222,"column":1}},"key":"yjscNU0eSx"},{"type":"text","value":"η","position":{"start":{"line":222,"column":1},"end":{"line":222,"column":1}},"key":"QtatwAm6Vu"},{"type":"text","value":" of the detector. (b) Logarithmic negativity defined in Eq. ","position":{"start":{"line":222,"column":1},"end":{"line":222,"column":1}},"key":"JpyisPncnf"},{"type":"crossReference","position":{"start":{"line":222,"column":1},"end":{"line":222,"column":1}},"children":[{"type":"text","value":"(","key":"e5LiLNyovn"},{"type":"text","value":"11","key":"QCfKVpe208"},{"type":"text","value":")","key":"zWRUmyndNR"}],"identifier":"lognegdefentang","label":"lognegdefentang","kind":"equation","template":"(%s)","enumerator":"11","resolved":true,"html_id":"lognegdefentang","key":"Mfjuryk2y3"},{"type":"text","value":", (c) initial thermal population and (d) squeezing parameter defined in Eq. 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The color and style of each curve match a different value of the second order correlation function, given in panel (a).","position":{"start":{"line":222,"column":1},"end":{"line":222,"column":1}},"key":"wVrnbAwJkD"}],"key":"FrET6RBrqG"}],"key":"CTTqgrGU7y"}],"enumerator":"6","html_id":"resume-entanglement","key":"UFt6LZzleV"},{"type":"paragraph","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"children":[{"type":"text","value":"Even more interesting is the initial thermal population of the state, shown in panel (c). 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Nonetheless, doing so, we are also counting thermal modes that contributes to increase the measured normalized variance.","position":{"start":{"line":248,"column":1},"end":{"line":248,"column":1}},"key":"XCDCkNqVLr"}],"key":"xsKhBbAfbB"},{"type":"comment","value":"A simple reason is due to the fact that averaging over many modes does not add noise since the statistics of an infinite number of mode is poissonian, their normalized variance ho In their case, they used a multimode source. As mode overlaps, the influence of the neighboring modes","position":{"start":{"line":249,"column":1},"end":{"line":249,"column":1}},"key":"LRags6uOYO"},{"type":"heading","depth":2,"position":{"start":{"line":253,"column":1},"end":{"line":253,"column":1}},"children":[{"type":"text","value":"Two-mode Fock probability distribution","position":{"start":{"line":253,"column":1},"end":{"line":253,"column":1}},"key":"bc6kP3QaCp"}],"identifier":"two-mode-fock-probability-distribution","label":"Two-mode Fock probability distribution","html_id":"two-mode-fock-probability-distribution","implicit":true,"enumerator":"5","key":"vBeLA37B68"},{"type":"paragraph","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"children":[{"type":"text","value":"Still assuming ","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"key":"rXBEWsdbEj"},{"type":"inlineMath","value":"d_k=0","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"html":"dk=0d_k=0dk​=0","key":"ZT7ALcj4aR"},{"type":"text","value":", the value of ","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"key":"fkAPwmbzh2"},{"type":"inlineMath","value":"g^{(2)}_{+-}","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"html":"g+−(2)g^{(2)}_{+-}g+−(2)​","key":"O4MfJwhECE"},{"type":"text","value":" and the population completely characterize the state. In particular, we can compute the expected two-mode probability distribution and compare it to experimental data. ","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"key":"m4Bw82kLhZ"},{"type":"crossReference","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"children":[{"type":"text","value":"Figure ","key":"CB0jhIXJnO"},{"type":"text","value":"7","key":"Zj30oBrFg9"}],"identifier":"vue_3d_reconstruit","label":"vue_3D_reconstruit","kind":"figure","template":"Figure %s","enumerator":"7","resolved":true,"html_id":"vue-3d-reconstruit","key":"Ntn0uzBjWv"},{"type":"text","value":" compares the measured state (left) and the expected one (right) given the measured populations and a correlation value of ","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"key":"qsklncZl8J"},{"type":"inlineMath","value":"g^{(2)}=2.2","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"html":"g(2)=2.2g^{(2)}=2.2g(2)=2.2","key":"SzBIlOP4U0"},{"type":"text","value":" (I choose 2.2 due to the last measurement of the variance). We observe a good agreement between the two distributions. From the quantum efficiency of the detector, we can even model the Fock probability distribution of the state. It is a two-mode squeezed thermal state with ","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"key":"IHiFx11WHc"},{"type":"inlineMath","value":"r_k\\sim 1","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"html":"rk∼1r_k\\sim 1rk​∼1","key":"Ho9SL6mBhT"},{"type":"text","value":" and ","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"key":"d8JduvE1Dt"},{"type":"inlineMath","value":"n_{th}^{(in)}=0.6","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"html":"nth(in)=0.6n_{th}^{(in)}=0.6nth(in)​=0.6","key":"SrBZqZ4zFJ"},{"type":"text","value":". It corresponds to the probability distribution that we would measure with a 100% quantum efficiency. We show this probability on the bottom panel of ","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"key":"mlOJOGE5Du"},{"type":"crossReference","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"children":[{"type":"text","value":"Figure ","key":"OmdHd5MIrh"},{"type":"text","value":"7","key":"gn5Jeq2gLP"}],"identifier":"vue_3d_reconstruit","label":"vue_3D_reconstruit","kind":"figure","template":"Figure %s","enumerator":"7","resolved":true,"html_id":"vue-3d-reconstruit","key":"LddzSIx3Dp"},{"type":"text","value":". On this probability distribution, we clearly show the pronounced diagonal which is a characteristic of a two-mode squeezed state. For a two-mode squeezed vacuum state, this two-mode probability distribution lies only on the diagonal. When there is a thermal seed, the diagonal is broadened.","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"key":"UCceqG2Ied"}],"key":"KwEQ9UVtx1"},{"type":"container","kind":"figure","identifier":"vue_3d_reconstruit","label":"vue_3D_reconstruit","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/vue_3D_reconstruit-2d6f5bfe67c579639af059b323276c6d.png","alt":"LogNeg","width":"100%","align":"center","key":"zTYFh8oklx","urlSource":"images/vue_3D_reconstruit.png","urlOptimized":"/~gondret/phd_manuscript/build/vue_3D_reconstruit-2d6f5bfe67c579639af059b323276c6d.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"vue_3D_reconstruit","identifier":"vue_3d_reconstruit","html_id":"vue-3d-reconstruit","enumerator":"7","children":[{"type":"text","value":"Figure ","key":"Z2QFsbdkBE"},{"type":"text","value":"7","key":"rSTW8COr9F"},{"type":"text","value":":","key":"T0MJSpGoN8"}],"template":"Figure %s:","key":"yE5KZlf7Uh"},{"type":"text","value":"Two-mode probability distribution as a function of the number of particles in mode 1 ","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"key":"f4krqu5LOC"},{"type":"inlineMath","value":"n_1","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"html":"n1n_1n1​","key":"Z1gcHfe1Qc"},{"type":"text","value":" and mode 2 ","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"key":"ffTOq8S7nn"},{"type":"inlineMath","value":"n_2","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"html":"n2n_2n2​","key":"UnztlqcG6b"},{"type":"text","value":". Left panel shows experimental data; Right panel shows the state model by a Gaussian TMSth state for which ","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"key":"fz3ULcAzlI"},{"type":"inlineMath","value":"g^{(2)}=2.2","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"html":"g(2)=2.2g^{(2)}=2.2g(2)=2.2","key":"vGT7Jwz4a7"},{"type":"text","value":" and a detected population of 0.92. Bottom panel shows the 2D distribution of the same state before detection (before the beam-splitter that models the losses). ©Theoretical distributions obtained using ","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"key":"WhDoGjAVCz"},{"type":"link","url":"https://the-walrus.readthedocs.io/en/latest/","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"children":[{"type":"text","value":"the Walrus","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"key":"g4zuRmkiAH"}],"urlSource":"https://the-walrus.readthedocs.io/en/latest/","key":"s1vRlGxbI6"},{"type":"text","value":" library ","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"key":"EatGr0nQ2k"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"children":[{"type":"cite","identifier":"gupt_walrus_2019","label":"gupt_walrus_2019","kind":"parenthetical","position":{"start":{"line":265,"column":500},"end":{"line":265,"column":517}},"children":[{"type":"text","value":"Gupt ","key":"vifrPy8H6L"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"lYMc59m8dh"}],"key":"V7C6DgxuFj"},{"type":"text","value":", 2019","key":"rZv6zyQprA"}],"enumerator":"5","key":"E2oNiRByWE"}],"key":"nfS1vJU5zB"},{"type":"text","value":".","position":{"start":{"line":265,"column":1},"end":{"line":265,"column":1}},"key":"GlhRz2M6dh"}],"key":"Lxh4hLwYpK"}],"key":"hlJYvYiQwD"}],"enumerator":"7","html_id":"vue-3d-reconstruit","key":"CfNg6cAMBc"},{"type":"heading","depth":2,"position":{"start":{"line":271,"column":1},"end":{"line":271,"column":1}},"children":[{"type":"text","value":"Conclusion","position":{"start":{"line":271,"column":1},"end":{"line":271,"column":1}},"key":"CBKqVZDM7p"}],"identifier":"conclu_ici","label":"conclu_ici","html_id":"conclu-ici","enumerator":"6","key":"WL3zmW2F02"},{"type":"paragraph","position":{"start":{"line":272,"column":1},"end":{"line":272,"column":1}},"children":[{"type":"text","value":"In this final section, we investigated deeper the single mode statistics to further check that it behaves like a thermal state. Such verification is important to use Wick theorem and use the theoretical work of chapter 2. In the second section we measured the population of the state. From the measured population and the value of the second order correlation function, we assess that the state is entangled. This is the major result of this PhD thesis.","position":{"start":{"line":272,"column":1},"end":{"line":272,"column":1}},"key":"LmGDs2jLwD"}],"key":"fObHwSxOgk"},{"type":"paragraph","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"children":[{"type":"text","value":"In the next section, we measured the fourth order correlation function. This measurement is compatible with a value ","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"key":"y5ixN876C0"},{"type":"inlineMath","value":"g^{(2)}=2.24(7)","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"html":"g(2)=2.24(7)g^{(2)}=2.24(7)g(2)=2.24(7)","key":"mXgJF77vOM"},{"type":"text","value":", even though the error-bars are quite large. This measurement is also slightly more compatible with the lower value of the ","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"key":"YAAYTXuHeG"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"RzEwWEbPzc"},{"type":"text","value":" uncertainty ","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"key":"Ol9ONKCV33"},{"type":"emphasis","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"key":"Hv03NPFGLz"}],"key":"SGcbW7ccaL"},{"type":"text","value":" 2.17. Finally, the 4-body correlation function is compatible with the fact that ","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"key":"rU3jf7XobD"},{"type":"inlineMath","value":"\\braket{\\hat{a}_k}=0","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"html":"⟨a^k⟩=0\\braket{\\hat{a}_k}=0⟨a^k​⟩=0","key":"XBoppgJXfQ"},{"type":"text","value":", which is expected from theory. In the last section, we further checked the consistency of our reasoning and, knowing the initial thermal population, we showed that our state is well-modeled by a two-mode squeezed thermal state with parameter ","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"key":"bFZJl7jsw6"},{"type":"inlineMath","value":"r_k=1.0","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"html":"rk=1.0r_k=1.0rk​=1.0","key":"xM9zWiRsmM"},{"type":"text","value":" and initial thermal population 0.6, detected with a 0.25 quantum efficiency. We also measured a normalized variance of 0.83(1). All measurements agree that","position":{"start":{"line":274,"column":1},"end":{"line":274,"column":1}},"key":"XNF7A1g068"}],"key":"C1Cj1aUWkA"},{"type":"list","ordered":true,"start":1,"spread":false,"position":{"start":{"line":275,"column":1},"end":{"line":279,"column":1}},"children":[{"type":"listItem","spread":true,"position":{"start":{"line":275,"column":1},"end":{"line":275,"column":1}},"children":[{"type":"text","value":"the state is entangled,","position":{"start":{"line":275,"column":1},"end":{"line":275,"column":1}},"key":"yMDcKxz95A"}],"key":"jyL8yiY5lS"},{"type":"listItem","spread":true,"position":{"start":{"line":276,"column":1},"end":{"line":276,"column":1}},"children":[{"type":"text","value":"the state that we detect has a logarithmic negativity of 0.4(1),","position":{"start":{"line":276,"column":1},"end":{"line":276,"column":1}},"key":"fvuE8wciGc"}],"key":"dcLn0RPDGG"},{"type":"listItem","spread":true,"position":{"start":{"line":277,"column":1},"end":{"line":279,"column":1}},"children":[{"type":"text","value":"if we take into account a 25(10)% quantum efficiency, the squeezing parameter of the reconstructed state is 1.0(7). The corresponding logarithmic negativity is 1.8(9).","position":{"start":{"line":277,"column":1},"end":{"line":277,"column":1}},"key":"uFT7uq8DPw"}],"key":"A6SXx1dwlJ"}],"key":"Gs3OEox9rY"},{"type":"paragraph","position":{"start":{"line":280,"column":1},"end":{"line":280,"column":1}},"children":[{"type":"text","value":"A future direction for this experiment is to monitor entanglement throughout the parametric creation process, ","position":{"start":{"line":280,"column":1},"end":{"line":280,"column":1}},"key":"XCxwZP6Tko"},{"type":"emphasis","position":{"start":{"line":280,"column":1},"end":{"line":280,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":280,"column":1},"end":{"line":280,"column":1}},"key":"zcuuWHIvKM"}],"key":"hh2SBG6Ixr"},{"type":"text","value":", by varying the excitation duration. Currently, we have not quantified entanglement through logarithmic negativity for each dataset as we have done in this chapter. However, we assessed the degree of violation of the classical Cauchy-Schwarz inequality (","position":{"start":{"line":280,"column":1},"end":{"line":280,"column":1}},"key":"FnrYrRAbND"},{"type":"inlineMath","value":"\\mathcal{C_S} > 1","position":{"start":{"line":280,"column":1},"end":{"line":280,"column":1}},"html":"CS>1\\mathcal{C_S} > 1CS​>1","key":"nEZ5a4K6VA"},{"type":"text","value":") and the relative number squeezing. These results are presented in ","position":{"start":{"line":280,"column":1},"end":{"line":280,"column":1}},"key":"RQm8C7U5Hg"},{"type":"crossReference","position":{"start":{"line":280,"column":1},"end":{"line":280,"column":1}},"children":[{"type":"text","value":"Figure","position":{"start":{"line":280,"column":1},"end":{"line":280,"column":1}},"key":"Camb7qKWzv"}],"identifier":"correlation_longue","label":"correlation_longue","kind":"figure","template":"Figure %s","enumerator":"8","resolved":true,"html_id":"correlation-longue","key":"GttB8y43Sv"},{"type":"text","value":" as a function of the detected population.","position":{"start":{"line":280,"column":1},"end":{"line":280,"column":1}},"key":"RpSxnfeM3L"}],"key":"GowRaJMTJd"},{"type":"container","kind":"figure","identifier":"correlation_longue","label":"correlation_longue","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/correlationn2-d3cf0a4edbe0117f0cb05094275057b5.png","alt":"correlation_longue","width":"100%","align":"center","key":"BWjoQTtzlY","urlSource":"images/correlationn2.png","urlOptimized":"/~gondret/phd_manuscript/build/correlationn2-d3cf0a4edbe0117f0cb05094275057b5.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":290,"column":1},"end":{"line":290,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"correlation_longue","identifier":"correlation_longue","html_id":"correlation-longue","enumerator":"8","children":[{"type":"text","value":"Figure ","key":"y4VDpBJlwK"},{"type":"text","value":"8","key":"gOOQ80olnb"},{"type":"text","value":":","key":"GVBMmlaKX4"}],"template":"Figure %s:","key":"Sh49oZg19Z"},{"type":"text","value":"Evolution of the normalized variance (right) and the Cauchy-Schwarz ratio (right) as a function of the detected population. The red horizontal line on the right subplot indicates the limit for relative number squeezing. On the right plot, the red lines show the bound on the threshold value for the Cauchy-Schwarz ratio derived in chapter 2. The dashed line does not take into account the quantum efficiency and the solid line account for a 25% quantum efficiency. ©Data taken in April 2024.","position":{"start":{"line":290,"column":1},"end":{"line":290,"column":1}},"key":"AI1bpNP6pg"}],"key":"NbVepu1X7V"}],"key":"KgmIrrUzHU"}],"enumerator":"8","html_id":"correlation-longue","key":"UuUxix6Zoz"},{"type":"paragraph","position":{"start":{"line":292,"column":1},"end":{"line":292,"column":1}},"children":[{"type":"text","value":"Assuming the state is Gaussian and thermal, the measurement of the Cauchy-Schwarz ratio and the population allow witnessing entanglement. Above the red line in ","position":{"start":{"line":292,"column":1},"end":{"line":292,"column":1}},"key":"mdJyVUeke2"},{"type":"crossReference","position":{"start":{"line":292,"column":1},"end":{"line":292,"column":1}},"children":[{"type":"text","value":"Figure ","key":"FNYYemwWn1"},{"type":"text","value":"8","key":"qhO5lHmzi5"}],"identifier":"correlation_longue","label":"correlation_longue","kind":"figure","template":"Figure %s","enumerator":"8","resolved":true,"html_id":"correlation-longue","key":"Te5v77LxSZ"},{"type":"text","value":", the state is entangled due to the lower bound on ","position":{"start":{"line":292,"column":1},"end":{"line":292,"column":1}},"key":"cwc71wdqNR"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":292,"column":1},"end":{"line":292,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"ZmsxukY7iT"},{"type":"text","value":" that we adapted for ","position":{"start":{"line":292,"column":1},"end":{"line":292,"column":1}},"key":"D3j3fsCglG"},{"type":"inlineMath","value":"\\mathcal{C_S}","position":{"start":{"line":292,"column":1},"end":{"line":292,"column":1}},"html":"CS\\mathcal{C_S}CS​","key":"eBTVs2p6NW"},{"type":"text","value":". The state is also ","position":{"start":{"line":292,"column":1},"end":{"line":292,"column":1}},"key":"Njkp0TRSGu"},{"type":"emphasis","position":{"start":{"line":292,"column":1},"end":{"line":292,"column":1}},"children":[{"type":"text","value":"particle entangled","position":{"start":{"line":292,"column":1},"end":{"line":292,"column":1}},"key":"LQx0sV6jAM"}],"key":"QT3lwEzQju"},{"type":"text","value":" without any hypothesis, as shown by ","position":{"start":{"line":292,"column":1},"end":{"line":292,"column":1}},"key":"zS3ycpkaJk"},{"type":"cite","identifier":"wasak_cauchy_schwarz_2014","label":"wasak_cauchy_schwarz_2014","kind":"narrative","position":{"start":{"line":292,"column":357},"end":{"line":292,"column":383}},"children":[{"type":"text","value":"Wasak ","key":"xEZP75bVZm"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"U2um6stvbe"}],"key":"EoVvEgN405"},{"type":"text","value":" (2014)","key":"YxiapZXOF4"}],"enumerator":"6","key":"G6Xa4zizkV"},{"type":"text","value":".","position":{"start":{"line":292,"column":1},"end":{"line":292,"column":1}},"key":"t04cEIYH2j"}],"key":"kQegs0NkOV"},{"type":"paragraph","position":{"start":{"line":294,"column":1},"end":{"line":294,"column":1}},"children":[{"type":"text","value":"These data were collected across various experimental configurations over different weeks. In ","position":{"start":{"line":294,"column":1},"end":{"line":294,"column":1}},"key":"OqtyUQ7oJN"},{"type":"crossReference","position":{"start":{"line":294,"column":1},"end":{"line":294,"column":1}},"children":[{"type":"text","value":"Figure","position":{"start":{"line":294,"column":1},"end":{"line":294,"column":1}},"key":"AdJdUMDk0S"}],"identifier":"correlationn","label":"correlationn","kind":"figure","template":"Figure %s","enumerator":"9","resolved":true,"html_id":"correlationn","key":"Xke7UeeFup"},{"type":"text","value":", we rescaled the different excitation processes to align them on a common ","position":{"start":{"line":294,"column":1},"end":{"line":294,"column":1}},"key":"mcem2iHfSu"},{"type":"inlineMath","value":"x","position":{"start":{"line":294,"column":1},"end":{"line":294,"column":1}},"html":"xxx","key":"GUe0xx1S9k"},{"type":"text","value":"-axis representing time. We used the dataset with filled markers in ","position":{"start":{"line":294,"column":1},"end":{"line":294,"column":1}},"key":"Q6J6Orjuj0"},{"type":"crossReference","position":{"start":{"line":294,"column":1},"end":{"line":294,"column":1}},"children":[{"type":"text","value":"Figure","position":{"start":{"line":294,"column":1},"end":{"line":294,"column":1}},"key":"pHSXjd3v17"}],"identifier":"correlationn","label":"correlationn","kind":"figure","template":"Figure %s","enumerator":"9","resolved":true,"html_id":"correlationn","key":"xRo5XZ5k0t"},{"type":"text","value":" as a reference. In this particular experiment, we excited the gas with a modulation amplitude of 8% for 8 periods at the 2 kHz breathing mode. After the excitation, we then let an additional delay ranging from 4 to 12 periods (2 to 6 ms), during which the ","key":"oC2t6kwA9c"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"kZ21cwpv35"}],"key":"bSH0wNx9yR"},{"type":"text","value":" continued breathing and excited pairs of quasi-particles. The time axis in ","key":"poIXjxYPSy"},{"type":"crossReference","position":{"start":{"line":294,"column":1},"end":{"line":294,"column":1}},"children":[{"type":"text","value":"Figure","position":{"start":{"line":294,"column":1},"end":{"line":294,"column":1}},"key":"EkEiM7ZPtM"}],"identifier":"correlationn","label":"correlationn","kind":"figure","template":"Figure %s","enumerator":"9","resolved":true,"html_id":"correlationn","key":"yM6ztU1wVQ"},{"type":"text","value":" thus refers to this additional delay. To align other experimental points, we calculated the theoretical growth rate based on the properties of each excitation process (amplitude, number of excitations, and additional delay) along with the specific ","key":"A7At2eXACK"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"i51kMTXQUt"}],"key":"L9Pb4l1Iwc"},{"type":"text","value":" properties of each dataset.","key":"rcZzdTwKsZ"}],"key":"X2LAhUzo7R"},{"type":"comment","value":"These data were taken in many experimental configurations on different weeks. In [](#correlationn), we rescaled the different excitation process to use the same $x$ axis as time. We use as a reference the dataset with the filled markers in [](#correlationn). For this experiment, we excited the gas with an amplitude of modulation of 8% for 8 periods at the 2 kHz breathing mode. We then scanned an additional duration from 4 to 12 periods (2 to 6 ms) during which the BEC keeps breathing and excites quasi-particles paris. The time in [](#correlationn) refers therefore to this additional excitation duration. 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From the ","position":{"start":{"line":315,"column":1},"end":{"line":315,"column":1}},"key":"n02sXlXSWy"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":315,"column":1},"end":{"line":315,"column":1}},"html":"g(2)g^{(2)}g(2)","key":"cxkMmKKHmi"},{"type":"text","value":" entanglement witness derived in the second chapter, we can already conclude on the non-separability of the state. 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