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The dissipation channels are considered explicitly in\norder to take advantage of the unitarity of the full scattering processes. We demonstrate that the\nCasimir energy is given by a scattering formula expressed in terms of the scattering amplitudes\ncoupling internal channels and taking dissipation into account implicitly. We prove that this formula\nis also valid when the surrounding medium is dissipative.\n\nKeywords: Casimir effect; scattering theory; dissipation\n\n1. Introduction\n\nCasimir physics has seen a renewed interest in recent decades thanks to new measurements of\nthe Casimir interaction between macroscopic objects [1–3] with an improved precision [4–8], as well\nas efforts to meet the associated theoretical challenges [9–14]. In order to accurately reproduce the\nexperimental data, a theoretical calculation has to model the optical properties of the materials used.\nA puzzling result of these comparisons is that some of the most precise experiments appear to agree\nwell with the calculations only when the Ohmic losses in the metallic plates are neglected in the\nmodel. Several possible explanations of this puzzle have been discussed, but none of them seem to be\nsatisfactory (a recent review is presented in [15]). For example, the electrostatic interaction between\npatches on the plates is certainly a possible systematic effect for Casimir force measurements [16–19],\nbut it does not explain the discrepancy between theory and measurements [20,21].\n\nThis still-unsolved discrepancy between experiment and theory has led to discussions about the\ncorrectness of the theoretical formula used to describe Casimir interaction. In particular, it has been\nrecently realized [22–24] that the calculations using the lossless plasma model were in fact neglecting\nthe interaction between magnetically coupled induced currents due to a subtlety in the mathematical\ndescription of causality properties of the metallic optical response. Though it does not solve the\ndiscrepancy, this work has shed interesting new light on the derivation of the scattering formula used\nin most calculations. Among other worries, it has also been suggested that the scattering approach\nmight not be valid for the dissipative metallic plates used in the experiments [25,26]. Some works have\nbeen devoted to ab initio treatments of the Casimir interaction between dissipative mirrors [27–30].\n\nIn the present article, we show that dissipation is taken into account in the usual scattering\nformula of the Casimir interaction energy [31,32]. We explicitly consider the channels responsible\nfor dissipation in order to take advantage of the unitarity of the scattering processes. In the end, the\nCasimir energy is given by a scattering formula written in terms of the scattering amplitudes of the\nmirrors, implicitly accounting for the channels responsible for dissipation. In the context of Casimir\n\nSymmetry 2018, 10, 37; doi:10.3390/sym10020037 www.mdpi.com/journal/symmetry\n\nhttp://www.mdpi.com/journal/symmetry\nhttp://www.mdpi.com\nhttps://orcid.org/0000-0003-4851-2198\nhttps://orcid.org/0000-0002-5193-1222\nhttps://orcid.org/0000-0002-1494-696X\nhttp://dx.doi.org/10.3390/sym10020037\nhttp://www.mdpi.com/journal/symmetry\n\n\nSymmetry 2018, 10, 37 2 of 11\n\nphysics, this result was already proven for the particular case of the plane-plane geometry [33,34],\nand the derivation in the present paper can be considered as a generalization to the case of an arbitrary\ngeometry. In a broader context, it is reminiscent of properties known in the theory of resistance in\nmesoscopic physics [35], or that of quantum field propagation in a dissipative medium [36,37].\n\n2. Scattering Interpretation of the Casimir Effect\n\nOur starting point is the interpretation of the Casimir effect in terms of the scattering\nformula [31,32]. Since temperature does not play a key role in the considerations presented below,\nwe assume T = 0 for the sake of simplicity. We begin by considering a single object placed into\na medium, with scattering of electromagnetic fluctuations by this object leading to a change of the\nvacuum energy written in terms of its scattering matrix S\n\n∆Evac = −h̄\nˆ ∞\n\n0\n\ndω\n\n2π\n∆φ , (1)\n\n∆φ =\n1\n2i\n\nlog det S . (2)\n\nThe change of vacuum energy ∆Evac is infinite when calculated for a single object, but its relevant\npart for estimating the Casimir effect turns out to be finite [38–40]. The phase shift ∆φ is the trace\nof eigen-phase shifts summed over all scattering channels at a given frequency ω. The Formula (1)\nthus has a clear physical meaning when the scattering matrix S is unitary, as it should if all scattering\nchannels are taken into account. Accordingly, it is obvious that ∆Evac is real.\n\nThis discussion does not mean that (1) cannot be applied when dissipation enters the game.\nIt only implies that all scattering channels responsible for dissipation processes must be included in the\nscattering theory. This can always be achieved, and necessarily leads to a unitary matrix. The general\nexpression (1) always describes the modification of the vacuum energy due to the presence of scatterers.\nAnother way to see that is to transform Equation (1) into an equivalent equation through an integration\nby parts and a rearrangement of terms:\n\n∆Evac =\n\nˆ ∞\n\n0\ndω\n\nh̄ω\n\n2\n∆η , (3)\n\n∆η =\n1\nπ\n\n∂\n\n∂ω\n∆φ . (4)\n\nHere, h̄ω/2 describes the vacuum energy of one mode at frequency ω, while ∆η is the modification\nof the density of states due to the presence of the scatterer [40,41]. Here again, this interpretation of (4)\nhas a direct physical meaning when the scattering matrix is unitary.\n\nIn the following, we derive the expression for the Casimir interaction energy between two objects\n1 and 2 . The set-up is displayed in Figure 1, with wavy lines representing dissipative channels for\nthe objects and the medium. We apply the formula written above for the total scattering matrix S\nviewed as describing the change of the electromagnetic vacuum energy when two objects are placed\nin the surrounding medium at a distance L. As depicted in Figure 1, the total scattering matrix S\ncan be decomposed into the scattering matrices S1 and S2 related to the individual objects, and the\nmatrix SL describing the propagation between the two objects over a distance L through the medium.\nThe expression for the Casimir interaction energy is then obtained as the change in the vacuum energy\ncaused by the full scattering matrix S after extracting the part depending on the distance L.\n\n\n\nSymmetry 2018, 10, 37 3 of 11\n\n1\n\nS1\n\n2\n\nS2\n\nL\n\nSL\n\nFigure 1. The Casimir interaction between two objects 1 and 2 at a distance L is considered.\nAs indicated by the wavy lines, both objects as well as the medium in between are generally\ndissipative. The two objects are described by unitary scattering matrices S1 and S2, which also\naccount for the external channels associated with the dissipation. The unitary scattering matrix SL\n\ndescribes the translation between the reference frames of objects 1 and 2 , and also accounts for the\nexternal channels.\n\nWe now introduce the notion of internal and external scattering channels. An internal scattering\nchannel links the two objects. It represents, for example, an outgoing channel from object 1 which\nbecomes an incoming channel at object 2 after propagation by a translation matrix, as discussed in\nSection 4. The channels which are not internal are named external channels. They account for the\nexchange of photons with the outside world and for quantum fluctuations from the environment. It is\nassumed that photons leaving through an external channel will not return coherently, but rather be\nabsorbed in an excitation process in the outside world. Once these channels are included, the scattering\nmatrices S1, S2, and SL are unitary, and therefore the total scattering matrix S is unitary as well.\nThe Casimir interaction is then given by the part of Equations (1) and (3), which depends on L. We\nshow below that the Casimir energy can also be described by a simplified scattering formula written\nin terms of scattering amplitudes between internal channels only, with the channels responsible for\ndissipation taken into account implicitly [33,34].\n\n3. Determinant Formula for Two Scatterers\n\nIn this section, we derive a relation involving determinants of scattering matrices for a scattering\nset-up with an internal structure described by two scattering matrices as depicted in Figure 2. In order\nto emphasize that the involved scattering matrices are general and not necessarily related to the\nscattering matrices introduced in Figure 1, we denote them by calligraphic symbols S , S1, and S2.\nWhen applying the relation for the determinant (17) obtained at the end of this section, we will replace\nthese general scattering matrices by specific scattering matrices related to the set-up shown in Figure 1.\n\nIgnoring the internal structure, the scattering properties can be described by a scattering matrix\nS coupling the ne\n\n1 + ne\n2 external channels among each other. Accounting for the internal structure,\n\nin addition to the ne\n1 and ne\n\n2 external channels associated with the scattering matrices S1 and S2,\nrespectively, one has ni internal channels coupling the two scatterers. Even though the two scatterers\nin Figure 2 are drawn at a certain distance, for the purpose of this section, we do not imply any effects\nof translation between the two scatterers. Such effects can be accounted for by an additional scattering\nmatrix, as we will see in Section 4.\n\n\n\nSymmetry 2018, 10, 37 4 of 11\n\nS\n\nne\n2 external channelsne\n\n1 external channels\n\nS1 S2\n\nni internal channels\n\nFigure 2. Scattering geometry with internal structure. Seen from the outside, a total of ne\n1 + ne\n\n2 external\nchannels are coupled by a scattering matrix S . The internal structure is accounted for by two scattering\nmatrices S1 and S2 coupling ni internal channels to ne\n\n1 and ne\n2 external channels, respectively.\n\nAs the individual scattering matrices S1 and S2 couple internal (i) and external (e) channels\namong each other, we can express them in block matrix form as\n\nSk =\n\n(\nS ii\n\nk S ie\nk\n\nSei\nk See\n\nk\n\n)\nk = 1, 2 . (5)\n\nThe global scattering matrix S is obtained by chaining the effect of the two individual scatterers\n\nS = S1 ? S2 , (6)\n\nwhere the symbol ? indicates that S is not obtained by a simple matrix multiplication of S1 and\nS2. In fact, the scattering matrices can be transformed into transfer matrices for which the chaining\ncorresponds to a matrix multiplication [33]. From the resulting transfer matrix, one obtains the global\nscattering matrix, which can be expressed as a block matrix\n\nS =\n\n(\nS11 S12\n\nS21 S22\n\n)\n, (7)\n\nwhere the blocks refer to the external channels associated with scatterers 1 and 2. Evaluating the\nchaining operation on S1 and S2 as just described, one finds\n\nS11 = See\n1 + Sei\n\n1 S ii\n2D21S ie\n\n1 (8a)\n\nS12 = Sei\n1 D12S ie\n\n2 (8b)\n\nS21 = Sei\n2 D21S ie\n\n1 (8c)\n\nS22 = See\n2 + Sei\n\n2 S ii\n1D12S ie\n\n2 , (8d)\n\nwhere\n\nD12 =\n(\n\n1− S ii\n2S ii\n\n1\n\n)−1\n(9a)\n\nD21 =\n(\n\n1− S ii\n1S ii\n\n2\n\n)−1\n. (9b)\n\nThe matrices in (9) account for an arbitrary number of round trips along the internal channels\nbetween the two scatterers starting on scatterer 1 and scatterer 2, respectively, as can be seen by means\nof a Taylor expansion; e.g.,\n\nD12 = 1 + S ii\n2S ii\n\n1 + S ii\n2S ii\n\n1S ii\n2S ii\n\n1 + S ii\n2S ii\n\n1S ii\n2S ii\n\n1S ii\n2S ii\n\n1 + · · · (10)\n\nThe relations (8a) and (8c) are visualized in Figure 3, and the other relations are obtained by\ninterchanging the two scatterers.\n\n\n\nSymmetry 2018, 10, 37 5 of 11\n\nS ii\n1\n\nS ie\n1\n\nSee\n1\n\nSei\n1\n\nS ii\n2\n\nS ie\n2\n\nSee\n2\n\nSei\n2\n\n(a)\n\ninternal channels\nex\nte\nrn\na\nl\nch\na\nn\nn\nel\ns ex\n\ntern\na\nl\nch\na\nn\nn\nels\n\nS11 block\n\nS ii\n1\n\nS ie\n1\n\nSee\n1\n\nSei\n1\n\nS ii\n2\n\nS ie\n2\n\nSee\n2\n\nSei\n2\n\n(b)\n\ninternal channels\n\nex\nte\nrn\na\nl\nch\na\nn\nn\nel\ns ex\n\ntern\na\nl\nch\na\nn\nn\nels\n\nS21 block\n\nFigure 3. Schematic representation of the blocks (a) S11 and (b) S21 of the total scattering matrix\nS = S1 ? S2. The diagrams visualize the Equations (8a) and (8c), respectively. The two other blocks\ndefined in (8b) and (8d) are obtained by exchanging the two objects.\n\nRelations (7)–(9) allow us to determine the determinant of the scattering matrix S . In the\nderivation, we suppose that the three matrices S , S1, and S2 are unitary. From the property (A7) of\nthe determinant of a unitary 2× 2 block matrix, we get together with the relations (8a) and (8d)\n\ndetS =\ndet(S22)\n\ndet(S†\n11)\n\n=\ndet(See\n\n2 + Sei\n2 S ii\n\n1D12S ie\n2 )\n\ndet(See\n1 + Sei\n\n1 S ii\n2D21S ie\n\n1 )\n∗\n\n. (11)\n\nThen, we use a generalization of the matrix determinant lemma on the above expression (see the\nAppendix A). For instance, for the numerator we have according to (A5)\n\ndet(See\n2 + Sei\n\n2 S ii\n1D12S ie\n\n2 ) = det(See\n2 )det(D12)det(D−1\n\n12 + S ie\n2 See\n\n2\n−1Sei\n\n2 S ii\n1 ) . (12)\n\nBy applying (A7) to matrices S1 and S2, we can express the determinants of the blocks See\n1 and\n\nSee\n2 related to the external channels by those related to the internal channels, S ii\n\n1 and S ii\n2 , and obtain\n\ndetS = det(S1)det(S2)\ndet(D12)\n\ndet(D21)∗\nα , (13)\n\nwhere the last factor reads\n\nα =\ndet(S ii\n\n2 )\n∗ det(D−1\n\n12 + S ie\n2 See\n\n2\n−1Sei\n\n2 S ii\n1 )\n\ndet(S ii\n1 )det(D−1\n\n21 + S ie\n1 See\n\n1\n−1Sei\n\n1 S ii\n2 )\n∗\n\n. (14)\n\n\n\nSymmetry 2018, 10, 37 6 of 11\n\nThis factor can be further evaluated by making use of (A8) yielding\n\nS ie\n1 See\n\n1\n−1Sei\n\n1 = S ii\n1 − S ii\n\n1\n†−1\n\n(15a)\n\nS ie\n2 See\n\n2\n−1Sei\n\n2 = S ii\n2 − S ii\n\n2\n†−1\n\n. (15b)\n\nEmploying those expressions and the definitions (9a) and (9b), we find that\n\nα =\ndet(S ii\n\n2\n† − S ii\n\n1 )\n\ndet(S ii\n1 − S ii\n\n2\n†\n)\n= (−1)ni\n\n(16)\n\nis only a phase factor depending on the number ni of internal channels. Finally, Sylvester’s determinant\nidentity implies detD12 = detD21, so that we get from (13) our first main result\n\ndetS = det(S1 ? S2) = (−1)ni\ndet(S1)det(S2)\n\ndet(D21)\n\ndet(D21)∗\n. (17)\n\n4. Application to the Casimir Interaction Energy\n\nAt first sight, it might appear that the result (17) can be directly applied to the expression for the\nCasimir energy (1) between two dissipative objects by replacing the general scattering matrices S1\n\nand S2 in (17) by the scattering matrices S1 and S2 of the two dissipative objects. However, as already\npointed out in the first paragraph of Section 3, the translation of the electromagnetic waves through\na potentially dissipative medium between the two objects has not yet been accounted for. Actually, we\nhave to consider the set-up depicted in Figure 4, where in addition to the scattering matrices S1 and\nS2, a scattering matrix SL is present. This scattering matrix describes the translation of electromagnetic\nwaves between the bases associated with objects 1 and 2 over a distance L. As shown in Figure 4,\nthe operator SL involves a first set of internal channels connecting S1 to SL, a second set of internal\nchannels connecting S2 to SL, and a set of external channels. The operator SL then has the same\nstructure as in Equation (5). However, using the two sets of internal channels defined previously,\nthe block Sii\n\nL itself has the following sub-structure:\n\nSii\nL =\n\n(\n0 Tii\n\n21\nTii\n\n12 0\n\n)\n. (18)\n\nAbove, the vanishing blocks express the fact that no backscattering can occur during the\npropagation between the two objects. The blocks Tii express the translation over a distance L from\nobject 1 to object 2 , and vice-versa. Concrete examples will be discussed at the end of this section.\nFurthermore, SL couples to external channels describing the loss of photons and the influence of noise\nfrom the environment. Those losses are described by the blocks Sei\n\nL , and are directly responsible for the\nimaginary part of the intervening medium’s refractive index. The global scattering matrix associated\nwith Figure 4 reads\n\nS = S1 ? SL ? S2 . (19)\n\nIn the chaining of scattering matrices, we are free to choose the order. As indicated by the box\nmarked by a dashed line in Figure 4, we start by evaluating SL ? S2.\n\n\n\nSymmetry 2018, 10, 37 7 of 11\n\nS = S1 ? SL ? S2\n\nSL ? S2\n\nS1 S2SL\n\nFigure 4. Set-up required to describe the Casimir effect. Apart from the scattering matrices S1 and\nS2, a scattering matrix SL describing the translation over a distance L is needed. In addition to the\ninternal channels, all scattering matrices also couple to external channels, thus allowing the dissipation\nof the objects and the medium in between to be accounted for. In a first step, the combination SL ? S2\n\nindicated by the dashed box is considered.\n\nWith SL and S2 being unitary matrices, we can directly apply (17) by replacing S1 and S2 by\nSL and S2, respectively. However, D21 reflecting the internal round-trips requires some attention.\nIn contrast to Section 3, the internal channels between scattering matrices S1 and S2 are now interrupted\nby the scattering matrix SL, and we should consider as internal only those channels connecting S2\n\nand SL. In contrast, the channels connecting S1 and SL are to be taken as external for the present\nconsideration. Since SL does not induce backscattering, it follows that the purely internal part of SL\nvanishes, Sii\n\nL = 0. As a consequence, D21 is a unit matrix, reflecting the fact that no internal round\ntrips are possible between SL and S2. From (17), we then obtain\n\ndet(SL ? S2) = (−1)ni\ndet(SL)det(S2) . (20)\n\nIn a second step, we apply (17) with S1 and S2 replaced by S1 and SL ? S2, and find together\nwith (20)\n\ndet S = det(S1)det(S2)det(SL)\ndet(D21)\n\ndet(D21)∗\n. (21)\n\nApart from S ii\n1 = Sii\n\n1 , the matrix D21 also contains the coupling between the internal channels\ndue to reflection by the chain of scattering matrices SL ? S2. As explained before, SL does not by\nitself lead to a coupling of internal channels linked to object 1 . This can happen only by means of Sii\n\n2\nsandwiched between translation matrices Tii\n\n12 and Tii\n21 through a dissipative medium over the distance\n\nL from object 1 to object 2 and back. In the last factor of (21), we thus have to set\n\nD21 =\n(\n\n1− Sii\n1 Tii\n\n12Sii\n2 Tii\n\n21\n\n)−1\n. (22)\n\nWe note that in the presence of a dissipative medium, Tii\n12 and Tii\n\n21 are non-unitary matrices.\nWe can now insert (21) together with (22) into (1) to obtain the change in the vacuum energy due\n\nto the dissipative scatterers separated by a dissipative medium. To obtain the Casimir interaction\nenergy, we need to identify the part which depends on the distance L between the two objects. In (21),\nthe first two factors depend only on properties of the individual objects, and are thus irrelevant for\nthe Casimir interaction energy. Only the last two factors depend on L. However, the global scattering\nmatrix S contains a trivial dependence on L arising from the shift of the basis discussed before (19).\nThis effect would survive even in the absence of the objects 1 and 2 , in which case the Casimir\n\n\n\nSymmetry 2018, 10, 37 8 of 11\n\ninteraction energy vanishes. We are thus left with the last factor. In view of (1) and (2), we finally\nobtain for the Casimir interaction energy\n\nECas(L) = h̄\nˆ ∞\n\n0\n\ndω\n\n2π\nIm log detD−1\n\n21\n\n= h̄\nˆ ∞\n\n0\n\ndω\n\n2π\nIm log det(1− Sii\n\n1 Tii\n12Sii\n\n2 Tii\n21) . (23)\n\nThis expression depends only on the parts of the scattering matrices pertaining to the internal\nchannels. Nevertheless, the properties of these parts reflect the dissipative properties of the objects\nand the medium in between.\n\nIn the form (23), the expression for the Casimir interaction energy is quite general and\nbasis-independent. The Dzyaloshinskii–Lifshitz–Pitaevskii formula [42] is recovered in the case\nof a plane-plane geometry. In this geometry, it makes sense to work in a basis of plane waves\ncharacterized by the quantum numbers {ω, q, ς}, where q = k − (k · L̂)L̂ is the transverse part\nof the wave vector k with respect to the unit vector L̂ normal to the two planes (note that q is a\nreal quantity since Im[k] is perpendicular to surfaces of constant amplitudes) and ς denotes the\npolarization. In this basis and this geometry, both the scattering matrix Sii and the translation\nmatrix Tii are diagonal, with matrix elements given by the Fresnel reflection amplitude r(ω, q, ς)\n\nand exp\n(\n\ni(n2ω2/c2 − q2)1/2L\n)\n\n, respectively. The lossy propagation is conveniently described by\nintroducing a complex refractive index n(ω) whose imaginary part is identified with the attenuation\nconstant. In the case of anisotropic intervening media, the refractive index becomes a tensor instead of\na scalar. Consequently, channels associated with different polarizations of the electromagnetic field\ncan be coupled through the scattering and translation operators Sii and Tii.\n\nAnother useful basis is the multipole basis {ω, `, m, ς} whenever the system under study has\nsome degree of spherical symmetry. For a sphere, the scattering matrix Sii is diagonal with elements\ndetermined by the Mie scattering amplitudes. The set of internal channels between a sphere and\nanother object consists of an infinite number of multipoles arising from translation formulas between\nspherical waves (see, e.g., [43]), so that the translation matrix Tii is not diagonal.\n\nFor geometries involving gratings (see, e.g., [44]), one works once again in a plane-wave basis.\nIn this case, it is the scattering matrix which is not diagonal due to the non-specular nature of the\nreflection by a grating. Therefore, the plane-plane geometry is one of the few examples where both\nscattering and translation matrices are diagonal (the other one being the somewhat unrealistic geometry\nconsisting of two concentric spheres). In general, at least one the two matrices is not diagonal. It is\npossible to treat in a similar way non-specular scattering for a Drude metal with Ohmic behaviour\nrelated to a disordered distribution of impurities [45]. Finally, we note that the formalism presented in\nthis manuscript can be generalized so as to include off-the-energy-shell scattering matrix elements in\norder to describe the dynamical Casimir effect observed in recent experiments [46–49].\n\n5. Conclusions\n\nWe have derived an expression for the Casimir interaction energy between dissipative objects\nembedded in a dissipative medium using the formalism of the scattering theory. The determinant of\nthe total scattering matrix can be factored out into parts depending or not on the distance between\nthe objects. The Casimir interaction energy is expressed using the distance-dependent part. Our final\nresult (23) depends exclusively on scattering matrix elements involving internal channels. Dissipation\nthus appears only implicitly in the scattering amplitudes, as the blocks over the internal channels are\nnon-unitary.\n\nAuthor Contributions: All authors contributed equally.\n\nConflicts of Interest: The authors declare no conflict of interest.\n\n\n\nSymmetry 2018, 10, 37 9 of 11\n\nAppendix A. Useful Lemmas\n\nIn this appendix, we gather several relations pertaining to block matrices which are required in\nthe main part of the text. Let\n\nM =\n\n(\nA B\nC D\n\n)\n(A1)\n\nbe a 2× 2 block matrix. Its determinant can expressed either as\n\ndet M = det(A)det(M/A) (A2)\n\nor\ndet M = det(D)det(M/D) , (A3)\n\nprovided that the blocks A and/or D are invertible. 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As indicated by the wavy lines, both objects as well as the medium in between are in general dissipative. The two objects are described by unitary scattering matrices $\\mathbf{S}_1$ and $\\mathbf{S}_2$ which account also for the external channels associated with the dissipation. The unitary scattering matrix $\\mathbf{S}_L$ describes the translation between the reference frames of objects \\protect\\circled{1} and \\protect\\circled{2} and also accounts for the external channels.","label":"fig:casimirSetup","source":"arxiv","key":"8fbea8dd9419770bf0954c6ae1308a92","url":"https://inspirehep.net/files/8fbea8dd9419770bf0954c6ae1308a92"},{"filename":"generalsetup.png","material":"preprint","caption":"Scattering geometry with internal structure. Seen from the outside, a total of $n^\\text{e}_1+n^\\text{e}_2$ external channels are coupled by a scattering matrix $\\bS$. The internal structure is accounted for by two scattering matrices $\\bS_1$ and $\\bS_2$ coupling $n^\\text{i}$ internal channels to $n^\\text{e}_1$ and $n^\\text{e}_2$ external channels, respectively.","label":"fig:generalSetup","source":"arxiv","key":"dfa8193a32ec0de6ab903826b8c3a450","url":"https://inspirehep.net/files/dfa8193a32ec0de6ab903826b8c3a450"},{"filename":"blocks.png","material":"preprint","caption":"Schematic representation of the blocks (a) $\\bS_{11}$ and (b) $\\bS_{21}$ of the total scattering matrix $\\bS = \\bS_{1} \\star \\bS_{2}$. The diagrams visualize the equations \\eqref{eq:blockOfS1} and \\eqref{eq:blockOfS3}, respectively. The two other blocks defined in \\eqref{eq:blockOfS2} and \\eqref{eq:blockOfS4} are obtained by exchanging the two objects.","label":"fig:scats","source":"arxiv","key":"16670c1571ba98a5f43cc6b0232107d4","url":"https://inspirehep.net/files/16670c1571ba98a5f43cc6b0232107d4"},{"filename":"translation.png","material":"preprint","caption":"Set-up required to describe the Casimir effect. Apart from the scattering matrices $\\mathbf{S}_1$ and $\\mathbf{S}_2$, a scattering matrix $\\mathbf{S}_L$ describing the translation over a distance $L$ is needed. In addition to the internal channels, all scattering matrices couple also to external channels, thus allowing to account for dissipation of the objects and the medium in between. In a first step, the combination $\\mathbf{S}_L\\star\\mathbf{S}_2$ indicated by the dashed box is considered.","label":"fig:translation","source":"arxiv","key":"adb4676fdee43e0e8886f6f9b32c9c0b","url":"https://inspirehep.net/files/adb4676fdee43e0e8886f6f9b32c9c0b"}],"legacy_version":"20190121090354.0","inspire_categories":[{"term":"General Physics","source":"arxiv"},{"term":"Quantum Physics","source":"arxiv"}],"first_author":{"emails":["guerout@lkb.upmc.fr"],"full_name":"Guérout, Romain","last_name":"Guérout","first_name":"Romain","recid":1921399},"control_number":1646335,"dois":[{"source":"bibmatch","value":"10.3390/sym10020037"},{"material":"publication","source":"arXiv","value":"10.3390/sym10020037"}],"document_type":["article"],"texkeys":["Guerout:2018mse","Guerout:2018oor"],"abstracts":[{"source":"MDPI","value":"We take dissipation into account in the derivation of the Casimir energy formula between two objects placed in a surrounding medium. The dissipation channels are considered explicitly in order to take advantage of the unitarity of the full scattering processes. We demonstrate that the Casimir energy is given by a scattering formula expressed in terms of the scattering amplitudes coupling internal channels and taking dissipation into account implicitly. We prove that this formula is also valid when the surrounding medium is dissipative.","abstract_source_suggest":{"input":"MDPI"}},{"source":"arXiv","value":"We take dissipation into account in the derivation of the Casimir energy formula between two objects placed in a surrounding medium. The dissipation channels are considered explicitly in order to take advantage of the unitarity of the full scattering processes. We demonstrate that the Casimir energy is given by a scattering formula expressed in terms of the scattering amplitudes coupling internal channels and taking dissipation into account in an implicit way. 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When the corrugation are supposed to be the smallest length scales, the effect of the corrugations can be studied in the perturbative expansion. This approximation will be dropped later on.","label":"lateral-fig1","source":"arxiv","key":"e30cc98d3a8d3026fd56446b62109ce2","url":"https://inspirehep.net/files/e30cc98d3a8d3026fd56446b62109ce2"},{"filename":"lateral-fig2.png","material":"preprint","caption":"Variation of $\\rho$ versus the dimensionless variable $\\kC L$ for metallic mirrors described by the plasma model, for $\\kP L=$1 (dashed line), 2.5 (dotted line), 5 (dashed-dotted line) and 10 (solid line) [colors online with respectively green, blue, red and black lines].","label":"lateral-fig2","source":"arxiv","key":"f8fd2bbad3c874eb84520f4dd4333d3b","url":"https://inspirehep.net/files/f8fd2bbad3c874eb84520f4dd4333d3b"},{"filename":"torque-fig1.png","material":"preprint","caption":"Casimir energy (arbitrary unit) as a function of the rotation angle $\\protect\\theta$ and the lateral displacement $b.$","label":"torque-fig1","source":"arxiv","key":"f19809569c8016c6e98d9f4285ed9742","url":"https://inspirehep.net/files/f19809569c8016c6e98d9f4285ed9742"},{"filename":"torque-fig2.png","material":"preprint","caption":"Maximum torque per unit area as a function of the mean separation $L$ for the following parameters: $a_1a_2=200\\,\\mathrm{nm}^2,$ $L_y=24\\,\\protect\\mu\\mathrm{m},$ $\\protect\\lambda_P=137\\, \\mathrm{nm}.$ Solid line: $\\protect\\lambda_C=2.4\\,\\protect\\mu\\mathrm{m};$ dashed line: $\\protect\\lambda_C=1.2\\,\\protect\\mu\\mathrm{m};$ dotted line: $ \\protect\\lambda_C=2\\protect\\pi L/2.6$ (corresponding to the optimum value).","label":"torque-fig2","source":"arxiv","key":"e5c579aa65ebb5c1b3cb31c6fb7ee79d","url":"https://inspirehep.net/files/e5c579aa65ebb5c1b3cb31c6fb7ee79d"},{"filename":"gratings-fig1.png","material":"preprint","caption":"Two surfaces with rectangluar gratings of depth $h$, gap width $d$ and trench width $d-d_1$.","label":"gratings-fig1","source":"arxiv","key":"39ed8d6ab431ee68be32832440b856af","url":"https://inspirehep.net/files/39ed8d6ab431ee68be32832440b856af"},{"filename":"gratings-fig2.png","material":"preprint","caption":"Casimir force normalized by its PFA value for two gratings of intrinsic Silicon with amplitude $h=100$nm and $d_1=\\frac{d}{2}$ as a function of $d$ at a fixed distance $L=250$nm.","label":"gratings-fig2","source":"arxiv","key":"949bed8312aa8e9498dc05c82fcc2ebc","url":"https://inspirehep.net/files/949bed8312aa8e9498dc05c82fcc2ebc"},{"filename":"ConvNDiffract.png","material":"preprint","caption":"Convergence of the calculated Casimir energy between two gratings as a function of the number of diffraction orders retained in the calculation. Gratings with different periods are plotted as blue and red ($400$nm) and green ($2\\ \\mu$m) points. The convergence of the calculations becomes slower when increasing the grating period $d$ or decreasing the separation $L$.","label":"fig:convN","source":"arxiv","key":"13dae8f2251d318f62811eb9e59ede81","url":"https://inspirehep.net/files/13dae8f2251d318f62811eb9e59ede81"},{"filename":"DeepGratings.png","material":"preprint","caption":"Comparisons between experimental measurements and exact calculations for the Casimir force gradient between a gold sphere and two types of silicon gratings. Green and red dots correspond to data points provided by Ho Bun Chan for a grating period of $400$nm and $1\\mu$m respectively. The solid curves of the same color are calculated data obtained using the scattering approach for the corresponding experimental parameters.","label":"fig:CompExp","source":"arxiv","key":"516a414040678590b5dea569286f547b","url":"https://inspirehep.net/files/516a414040678590b5dea569286f547b"},{"filename":"Plane-sphere.png","material":"preprint","caption":"The geometry of a sphere of radius $R$ and a plate at distance $L$; the center-to-plate distance is $\\cL\\equiv L+R$.","label":"Plane-sphere","source":"arxiv","key":"cfe4129688bdedd742581a4e3dfccc93","url":"https://inspirehep.net/files/cfe4129688bdedd742581a4e3dfccc93"},{"filename":"rho_E_LORs.png","material":"preprint","caption":"Upper graph: the ratio $\\rho_E={E}/{E^\\PFA}$ of the plane-sphere Casimir energy to its PFA estimation is plotted as a function of $\\ell_\\max$ for different values of $L/R=0.05, 0.1, 0.2$. Lower graph: same ratio $\\rho_E$ plotted as function of $L/R$ for different values of $\\ell_\\max=20,40,80$.","label":"EffectEllMax","source":"arxiv","key":"9042f0a654259710ada7cdcf8a84fbd1","url":"https://inspirehep.net/files/9042f0a654259710ada7cdcf8a84fbd1"},{"filename":"rho_E_T0_lmaxs.png","material":"preprint","caption":"Upper graph: the ratio $\\rho_E={E}/{E^\\PFA}$ of the plane-sphere Casimir energy to its PFA estimation is plotted as a function of $\\ell_\\max$ for different values of $L/R=0.05, 0.1, 0.2$. Lower graph: same ratio $\\rho_E$ plotted as function of $L/R$ for different values of $\\ell_\\max=20,40,80$.","label":"EffectEllMax","source":"arxiv","key":"8e6422c369048e55369b1c9de6637a50","url":"https://inspirehep.net/files/8e6422c369048e55369b1c9de6637a50"},{"filename":"plan-sphere-fig3.png","material":"preprint","caption":"Variation of $\\rho_\\G$ as a function of $L/R$ as a function of $L/R$, for a nanosphere of radius $R=100$nm; the solid green line corresponds to gold-covered plates ($\\lambda_\\P=136$nm) and the dashed red line to perfect reflectors. The decrease at low values of $L/R$ represent a numerical inaccuracy due to the limited value of $\\ell_\\max$ (24 in this calculation \\cite{CanaguierPRL09}).","label":"plane-sphere-fig2","source":"arxiv","key":"175e8835e22495d02409c313e02e0ed6","url":"https://inspirehep.net/files/175e8835e22495d02409c313e02e0ed6"},{"filename":"plane-sphere-T-fig1bis.png","material":"preprint","caption":"Thermal Casimir force at $T=300$K divided by the zero temperature force, computed between perfectly reflecting sphere and plane (upper graph), and between Drude metals (lower graph) plotted for $\\lambda_\\P=136$nm, $\\lambda_{\\gamma}/\\lambda_\\P=250$. The solid lines from bottom to top correspond to increasing values of sphere radii. The dotted curve in the upper graph is the analytical asymptotic expression in the $L \\gg R$ limit. The PFA expressions are given by the dashed curves.","label":"plane-sphere-T-fig1","source":"arxiv","key":"7a871a454774d85582255ba659e931cf","url":"https://inspirehep.net/files/7a871a454774d85582255ba659e931cf"},{"filename":"plane-sphere-T-fig2bis.png","material":"preprint","caption":"Thermal Casimir force at $T=300$K divided by the zero temperature force, computed between perfectly reflecting sphere and plane (upper graph), and between Drude metals (lower graph) plotted for $\\lambda_\\P=136$nm, $\\lambda_{\\gamma}/\\lambda_\\P=250$. The solid lines from bottom to top correspond to increasing values of sphere radii. The dotted curve in the upper graph is the analytical asymptotic expression in the $L \\gg R$ limit. The PFA expressions are given by the dashed curves.","label":"plane-sphere-T-fig1","source":"arxiv","key":"19011e37cb1b71e6de81eab9f00d1603","url":"https://inspirehep.net/files/19011e37cb1b71e6de81eab9f00d1603"},{"filename":"plane-sphere-T-fig3.png","material":"preprint","caption":"Ratio of thermal Casimir force at $T=300K$ calculated with the plasma model and the Drude model, as a function of surface separation $L$ for different radii of the sphere. The solid curves from bottom to top correspond to increasing values of sphere radii. The dashed curve is the PFA prediction.","label":"plane-sphere-T-fig3","source":"arxiv","key":"8cdb532c4c6810f7a3bc5b87bec74593","url":"https://inspirehep.net/files/8cdb532c4c6810f7a3bc5b87bec74593"}],"legacy_version":"20171124002448.0","inspire_categories":[{"term":"General Physics"},{"term":"Theory-HEP"},{"term":"Quantum Physics","source":"arxiv"}],"first_author":{"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/01h14ww21"}],"full_name":"Lambrecht, Astrid","last_name":"Lambrecht","first_name":"Astrid","recid":1001198},"control_number":1225058,"dois":[{"value":"10.1007/978-3-642-20288-9_4"},{"material":"publication","source":"arXiv","value":"10.1007/978-3-642-20288-9_4"}],"document_type":["book chapter"],"texkeys":["Lambrecht:2010qfa"],"abstracts":[{"source":"arXiv","value":"We present calculations of the quantum and thermal Casimir interaction between real mirrors in electromagnetic fields using the scattering approach. We begin with a pedagogical introduction of this approach in simple cases where the scattering is specular. We then discuss the more general case of stationary arbitrarily shaped mirrors and present in particular applications to two geometries of interest for experiments, that is corrugated plates and the plane-sphere geometry. The results nicely illustrate the rich correlations existing between material properties, temperature and geometry in the Casimir effect.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["quant-ph"],"titles":[{"source":"arXiv","title":"Casimir effect in the scattering approach: correlations between material properties, temperature and geometry"}],"curated":true},"id":"1225058","created":"2013-03-23T00:00:00+00:00"},{"updated":"2025-10-02T14:11:42.118004+00:00","links":{"bibtex":"https://inspirehep.net/api/literature/3025774?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/3025774?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/3025774?format=latex-us","json":"https://inspirehep.net/api/literature/3025774?format=json","json-expanded":"https://inspirehep.net/api/literature/3025774?format=json-expanded","cv":"https://inspirehep.net/api/literature/3025774?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A3025774"},"metadata":{"publication_info":[{"journal_volume":"115","pubinfo_freetext":"Phys. 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The time variable runs along the vertical.","label":"fig1","source":"arxiv","key":"fe32df746d4d86eadb497bf4f1ce6ab1","url":"https://inspirehep.net/files/fe32df746d4d86eadb497bf4f1ce6ab1"},{"filename":"Fig3.png","material":"preprint","caption":"Graphical representation of the interaction between a $^{87}$Rb atom in state $5P_{1/2}$ $(k_{A}=2\\pi\\:12578.95\\:$cm$^{-1})$ and a $^{40}$K atom in its ground state,  for $k_{A}R\\gg1$. The black line corresponds to a snapshot of the interaction at time $3.0\\cdot10^{-12}\\:$s, where the contributions of the  D1 and D2 transition lines of the  $^{40}$K atom ($k^{1}_{B}=2\\pi\\:12985.17\\:$cm$^{-1}$ and  $k^{2}_{B}=2\\pi\\:13045.876\\:$cm$^{-1}$ respectively) add up approximately in-phase. A long-range period $c\\pi/|\\bar{\\Delta}_{AB}|$, with $\\bar{\\Delta}_{AB}/c=k_{A}-(k_{B}^{1}+k_{B}^{2})/2$, is identified. The red line corresponds to the time-independent result of the causal-adiabatic approximation. Average over dipole orientations has been taken.","label":"fig3","source":"arxiv","key":"42563ea516109d29a631108fe681bd41","url":"https://inspirehep.net/files/42563ea516109d29a631108fe681bd41"},{"filename":"Fig2.png","material":"preprint","caption":"Dominant diagrams at order $\\mathcal{O}(W^{6})$ which incorporate  in the calculation of $\\langle W_{A}(T)\\rangle$ the effect of photon emission into free space. In $(i)$ and $(ii)$ the photon $\\gamma$ is emitted into free space from atom $A$, whereas in $(iii)$ it is emitted from atom $B$.","label":"fig2","source":"arxiv","key":"8d571007a2aa983c024931c13e07ba78","url":"https://inspirehep.net/files/8d571007a2aa983c024931c13e07ba78"}],"inspire_categories":[{"term":"Quantum Physics","source":"arxiv"}],"preprint_date":"2015-06-23","author_count":3,"first_author":{"full_name":"Donaire, Manuel","last_name":"Donaire","first_name":"Manuel","recid":1034169},"public_notes":[{"source":"arXiv","value":"Appendix added and minor corrections made to agree with the published version"}],"control_number":3025774,"dois":[{"material":"publication","source":"arXiv","value":"10.1103/PhysRevLett.115.033201"}],"earliest_date":"2015-06-23","document_type":["article"],"texkeys":["Donaire:2015bho"],"abstracts":[{"source":"arXiv","value":"We present a time-dependent quantum calculation of the van der Waals interaction between a pair of dissimilar atoms, one of which is initially excited while the other one is in its ground state. For small detuning, the interaction is predominantly mediated at all distances by the exchange of doubly resonant photons between the two atoms. We find that it presents both temporal and spatial oscillations. Spatially oscillating terms depend on the resonant frequencies of both atoms, while the frequency of the time oscillations is given by their detuning. We analyse the physical content of our findings and discuss to what extent previous conflicting stationary approaches provide compatible results. A proper account of causality is found essential in order to obtain the correct result.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["quant-ph"],"titles":[{"source":"arXiv","title":"Quasi-resonant van der Waals interaction between non-identical atoms"}],"facet_author_name":["1921399_Romain Gu´erout","1001198_Astrid Lambrecht","1034169_Manuel Donaire"],"license":[{"license":"arXiv nonexclusive-distrib 1.0","material":"preprint","url":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/"}],"_oai":{"sets":["Literature"],"id":"oai:inspirehep.net:3025774","updated":"2025-10-02T14:11:42.118004"},"curated":false,"journal_title_variants":["Phys.Rev.Lett.","Phys. Rev. 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The filling factor $p=p'/d$ is given as a percentage of the period $d$.","label":"fig:epsart1","source":"arxiv","key":"7e6215f931f0e903b8e0d0d4f3c9ed11","url":"https://inspirehep.net/files/7e6215f931f0e903b8e0d0d4f3c9ed11"},{"filename":"Distance.png","material":"preprint","caption":"Heat transfer coefficients as a function of separation distance $L$, when the gratings are not laterally displaced (blue solid line) and when they are by half a period (blue dashed line). This is compared with the Proximity Approximation in red. The gratings have a period $d=1500$nm, filling factor $p=20\\%$, and groove depth $a=500$nm.","label":"fig:epsart2","source":"arxiv","key":"134156017f0e5e7bf915c5326f4c0f0a","url":"https://inspirehep.net/files/134156017f0e5e7bf915c5326f4c0f0a"},{"filename":"FieldModulus.png","material":"preprint","caption":"Field modulus map of a given source dipole placed in the middle of a corrugation and right under the surface. The field is here represented only in the upper grating, so as to highlight where the absorption takes place. This is for gratings at a separation distance $L=25$nm. The two figures on the left display the profiles in the $xz-$plane (in green) when they are aligned ($\\delta=0$), and the two figures on the right when they are laterally displaced by half-a-period ($\\delta=d/2$), both for two different wavelengths $\\lambda=8.75 \\mu$m (top) and $9.15 \\mu$m (down).","label":"fig:epsart2b","source":"arxiv","key":"382ac83b81ef3905f0db328e7bc7c2f4","url":"https://inspirehep.net/files/382ac83b81ef3905f0db328e7bc7c2f4"},{"filename":"Planeplane.png","material":"preprint","caption":"Heat transfer coefficients as a function of the separation distance $L$ between two plane mirrors of SiO$_2$ (red solid curve), compared with the black body limit (blue dashed line). One can divide the separation distance in three domains A, B, and C, respectively corresponding to the extreme near-field below $200$nm, to the near-field from $200$nm to $10\\mu$m, and to the domain of Stefan-Boltzmann's law beyond $10\\mu$m. This can be seen by the change of the slope of the curve along these three ranges.","label":"fig:epsart3","source":"arxiv","key":"9c8e2ec11906bf3ee741a1e08eb7943f","url":"https://inspirehep.net/files/9c8e2ec11906bf3ee741a1e08eb7943f"},{"filename":"Period.png","material":"preprint","caption":"Heat transfer coefficients as a function of grating period $d$, when the gratings are not laterally displaced (solid blue line) and when they are displaced by half a period (dashed blue line). This is compared with the PA in red. The gratings have a groove depth $a=500$nm, filling factor $p=20\\%$, and are at a separation distance $L=100$nm.","label":"fig:epsart4","source":"arxiv","key":"45089685a35dd0338e7f90ce308503da","url":"https://inspirehep.net/files/45089685a35dd0338e7f90ce308503da"},{"filename":"Fillingfactor.png","material":"preprint","caption":"Heat transfer coefficients as a function of filling factor $p$, when the gratings are not laterally displaced (solid blue line) and when they are displaced by half a period (dashed blue line). Respective PA predictions are in red. The dotted gray line is the percentage of the modulation factor $h_{\\delta = 0} / h_{\\delta = d/2}$. Gratings have a period and groove depth of $500$nm, and are separated by a distance $L=100$ nm.","label":"fig:epsart5","source":"arxiv","key":"3d8c92a302d2dfe4f1a9ec20c11d8aee","url":"https://inspirehep.net/files/3d8c92a302d2dfe4f1a9ec20c11d8aee"}],"inspire_categories":[{"term":"Quantum Physics","source":"arxiv"},{"term":"Condensed Matter","source":"arxiv"}],"preprint_date":"2012-06-01","author_count":6,"first_author":{"full_name":"Lussange, J.","last_name":"Lussange","first_name":"J.","recid":2237258},"control_number":3013923,"dois":[{"material":"publication","source":"arXiv","value":"10.1103/PhysRevB.86.085432"}],"earliest_date":"2012-06-01","document_type":["article"],"texkeys":["Lussange:2012nbi"],"abstracts":[{"source":"arXiv","value":"We present a theoretical study of radiative heat transfer between dielectric nanogratings in the scattering approach. As a comparision with these exact results, we also evaluate the domain of validity of Derjaguin's Proximity Approximation (PA). We consider a system of two corrugated silica plates with various grating geometries, separation distances, and lateral displacement of the plates with respect to one another. Numerical computations show that while the PA is a good approximation for aligned gratings, it cannot be used when the gratings are laterally displaced. We illustrate this by a thermal modulator device for nanosystems based on such a displacement.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["quant-ph"],"titles":[{"source":"arXiv","title":"Radiative heat transfer between two dielectric nanogratings in the scattering approach"}],"facet_author_name":["1921399_Romain Gu´erout","1970087_Jean-Jacques Greffet","2237258_J. Lussange","989288_Serge Reynaud","1001198_Astrid Lambrecht","2398331_Felipe S.S. Rosa"],"license":[{"license":"arXiv nonexclusive-distrib 1.0","material":"preprint","url":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/"}],"_oai":{"sets":["Literature"],"id":"oai:inspirehep.net:3013923","updated":"2026-02-09T15:13:32.640371"},"curated":false,"journal_title_variants":["Phys.Rev.B","Phys. Rev. B"],"arxiv_eprints":[{"categories":["quant-ph","cond-mat.other"],"value":"1206.0211"}]},"id":"3013923","created":"2025-10-02T09:27:55.614358+00:00"},{"updated":"2025-11-10T08:24:59.461906+00:00","links":{"bibtex":"https://inspirehep.net/api/literature/1671220?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/1671220?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/1671220?format=latex-us","json":"https://inspirehep.net/api/literature/1671220?format=json","json-expanded":"https://inspirehep.net/api/literature/1671220?format=json-expanded","cv":"https://inspirehep.net/api/literature/1671220?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A1671220"},"metadata":{"publication_info":[{"page_end":"184","conference_record":{"$ref":"https://inspirehep.net/api/conferences/1497324"},"year":2017,"parent_isbn":"9791096879052","page_start":"179","parent_record":{"$ref":"https://inspirehep.net/api/literature/1639583"},"cnum":"C17-03-25"}],"authors":[{"full_name_unicode_normalized":"crépin, p-p","full_name":"Crépin, P-P","record":{"$ref":"https://inspirehep.net/api/authors/2185728"},"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/907500"},"value":"Paris, Lab. Kastler Brossel"}],"last_name":"Crépin","ids":[{"schema":"INSPIRE BAI","value":"P.P.Crepin.1"}],"signature_block":"CRAPANp","first_name":"P-P","uuid":"40004e2f-65bd-4bc6-8f3d-0ced6a6ca8d9","recid":2185728},{"full_name_unicode_normalized":"dufour, g.","full_name":"Dufour, G.","record":{"$ref":"https://inspirehep.net/api/authors/2152325"},"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/907500"},"value":"Paris, Lab. Kastler Brossel"}],"last_name":"Dufour","ids":[{"schema":"INSPIRE BAI","value":"G.Dufour.2"}],"signature_block":"DAFARg","first_name":"G.","uuid":"84ae1c4f-5460-41e9-b62b-5c2928346aa1","recid":2152325},{"full_name_unicode_normalized":"guérout, r.","full_name":"Guérout, R.","record":{"$ref":"https://inspirehep.net/api/authors/1921399"},"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/907500"},"value":"Paris, Lab. Kastler Brossel"}],"last_name":"Guérout","ids":[{"schema":"INSPIRE BAI","value":"R.Guerout.1"}],"signature_block":"GARATr","first_name":"R.","uuid":"159aa5cf-c066-4f70-a5dc-179c565f6db7","recid":1921399},{"full_name_unicode_normalized":"lambrecht, a.","full_name":"Lambrecht, A.","record":{"$ref":"https://inspirehep.net/api/authors/1001198"},"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/907500"},"value":"Paris, Lab. Kastler Brossel"}],"last_name":"Lambrecht","ids":[{"schema":"INSPIRE BAI","value":"A.Lambrecht.1"}],"signature_block":"LANBRACHTa","first_name":"A.","uuid":"086d1198-042e-44ef-a2ec-aa4516679584","recid":1001198},{"full_name_unicode_normalized":"reynaud, s.","full_name":"Reynaud, S.","record":{"$ref":"https://inspirehep.net/api/authors/989288"},"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/907500"},"value":"Paris, Lab. Kastler Brossel"}],"last_name":"Reynaud","ids":[{"schema":"INSPIRE BAI","value":"S.Reynaud.1"}],"signature_block":"RYNADs","first_name":"S.","uuid":"5cd54b41-f1ad-406f-aa5a-d11536964574","recid":989288}],"citation_count_without_self_citations":0,"citation_count":0,"documents":[{"filename":"1639583_179-184.pdf","attachment":{"content":"Casimir-Polder shifts on quantum levitation states\n\nP-P. Crépin, G. Dufoura, R. Guérout, A. Lambrecht and S. Reynaud\nLaboratoire Kastler Brossel, UPMC-Sorbonne Université, CNRS, ENS-PSL Research University,\n\nCollège de France, Campus Jussieu case 74, F-75252 Paris, France.\n\nUltracold atoms can be held in quantum levitation states by the joint effect of gravity and\nquantum reflection from the attractive Casimir-Polder interaction above a horizontal mirror.\nWe calculate the Casimir-Polder shifts of the energies of these states with a precision sufficient\nto discuss spectroscopy experiments aiming at tests of the weak equivalence principle on\nantihydrogen.\n\nGravitationally bound quantum states have been observed with ultracold neutrons 1,2, but\nnot with neutral atoms. At the low energies corresponding to such bound states, an efficient\nquantum reflection is produced by the Casimir-Polder (CP) tail above a surface 3,4,5,6, so that\nquantum levitation states of atoms can be created, where quantum reflection above a horizontal\nmirror balances gravity 7.\n\nIt has recently be proposed to test the weak equivalence principle with antihydrogen by\ntiming its free fall from a trapping device 8. The precision of this test, of the order of 1%, could\nbe improved by spectroscopic measurements of quantum levitation states of antihydrogen above\na material surface, since the properties of these states are essentially determined by gravity 9,10.\nHowever an accurate determination of the acceleration requires a precise evaluation of Casimir-\nPolder shifts on quantum levitation states 11.\n\nWe consider an atom of mass m and energy E bouncing in the gravity potential mgz and\nreflected on a plane horizontal mirror due to quantum reflection on the CP potential VCP(z).\nThe wavefunction ψ(z) obeys the one-dimensional Schrödinger equation\n\n− ~2\n\n2m\n\nd2ψ\n\ndz2\n(z) + V (z)ψ(z) = Eψ(z) , V (z) = mgz + VCP(z) , (1)\n\nwith the potential V depending only on the altitude z of the particle above the mirror sitting at\nz = 0. We also suppose that atoms are absorbed when touching the surface, which corresponds\nto the physical boundary condition for antihydrogen annihilated at contact with matter.\n\nLength and energy scales associated with the Casimir-Polder potential are respectively much\nsmaller and much larger than those associated with the gravitational potential\n\n`g =\n\n(\n~2\n\n2m2g\n\n)1/3\n\n≈ 5.87 µm , εg = mg`g ≈ 0.602 peV , (for g ≈ 9.81 m.s−2) . (2)\n\nAn approximate solution of the problem is thus found by decoupling the effects of gravity and\nCP interactions. For quantum levitation states with low values of the quantum number n, the\nscattering amplitudes are mainly given by the scattering length a. It follows that the energies E0\n\nn\n\naPhysikalisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104, Freiburg, Germany\n\n\n\nof an ideal quantum bouncer are all shifted by the same quantity mga resulting from the complex\nphase shift experienced by the atom upon reflection on the CP tail 9. Within this scattering\nlength approximation where energies are E1n = E0\n\nn + mga, the transition frequencies between\nquantum states are independent of the atom-surface interaction. Spectroscopy experiments\nthus give a direct access to the value of εg, that is also g. In the following, we present an exact\ntreatment of the full potential including the effects of gravity and CP interaction, which allows\nus to assess the accuracy of this approximation. We also give improved numerical and analytical\nresults sufficient for discussing the proposed spectroscopic tests of free fall 9,10.\n\nThe Schrödinger equation, rewritten in the following form\n\nψ′′(z) + F (z)ψ(z) = 0 , F (z) = kdB(z)2 =\n2m\n\n~2\n(E − V (z)) , (3)\n\nwith F (z) the square of the de Broglie wavevector kdB can be transformed by a Liouville trans-\nformation consisting in a coordinate change z → z associated with a rescaling ψ → ψψψ of the\nwave-function 12. The transformed wave function obeys a Schrödinger equation (3) with the\ntransformed F−function written in terms of a transformed energy E and potential V . Figure\n1 shows the original potential landscape and transformed one. The potential V is the sum of\na linear gravity potential and an effective potential VCP(z) producing quantum reflection. In\nsharp contrast with the CP well on Figure 1, the transformed potential VCP now shows a high\npeak close to the surface.\n\n0 1 2 3 4 5 6 7 8\nz (`g)\n\n−2\n\n0\n\n2\n\n4\n\n6\n\n8\n\nV\n,E\n\n(ε\ng\n)\n\n0.00 0.05 0.10\n\n−20\n\n−10\n\n0\n\n−2 0 2 4 6 8\nz\n\n−2\n\n0\n\n2\n\n4\n\n6\n\n8\n\nV\n,\nE\n\n−0.1 0.0 0.1 0.2\n\n0\n\n200\n\n400\n\n600\n\nFigure 1 – Left plot: the landscape for antihydrogen atom in gravity and CP potentials above a silica bulk (black\ncurve). Horizontal lines correspond to energies chosen as E0\n\nn (n = 1, ..., 5 for the blue, green, red, cyan and yellow\nlines, respectively from bottom to top line). A zoom on the potential well near the surface is shown in the inset.\nRight plot: the same problem after a Liouville transformation, with energies chosen as E0\n\nn(same color and position\ncodes as for the left plot) and potential V = z + VCP. A zoom on the wall is shown in the inset.\n\nWith quantum reflection now understood as classically expected reflection on a repulsive\nwall, we get a new physical picture for quantum levitation states corresponding to matter waves\ntrapped in a Fabry-Perot cavity. Gravity plays the role of the vertical cavity’s perfectly reflective\ntop mirror, while quantum reflection corresponds to its partially reflective bottom mirror. We\ncan then interpret the properties of quantum levitation states in terms of cavity resonances.\n\nAbove the top mirror, the solution of the Schrödinger equation is given by Airy functions,\nwhich are linear superpositions of upward and downward traveling waves Ci+ and Ci− (zt is the\nposition of the classical turning point in the transformed frame)\n\nψψψm(z) =\nam\n2\n\n(\nCi+(z − zt) + Ci−(z − zt)\n\n)\n, Ci±(z) = Ai(z)± iBi(z) . (4)\n\nThe amplitude am depends on the number m of bounces of the matter wave on the bottom\nmirror. With the ideal quantum bouncer model, the ideal energy levels E0\n\nn would be recovered\n\n\n\nby obtaining the stationary quantum solutions of (4). The more general problem studied in this\npaper is not unitary since atoms transmitted through the bottom mirror are lost.\n\nAs a consequence, the quantum levitation states can only be obtained as quasi-stationary\nstates, with the amplitude am decreasing after each bounce, due to the losses. In analogy with\nthe theory of optical Fabry-Perot cavities, we introduce a factor describing the modification of\nthe traveling waves after one cavity round trip\n\nam+1 = ρam , (5)\n\nwith ρ obtained by solving numerically the quantum reflection problem. We may define energies\nEn by requiring ρ(En) to be a real number slightly smaller than unity. The value attained for\nρ(En) is related to the loss at each bounce, that is also the finesse of the cavity resonance. Figure\n2 shows energy shifts En−E0\n\nn, which are close to the constant value mgRe(a) predicted by the\nscattering length approximation discussed above with an accuracy of the order of 10−4.\n\n●\n● ● ●\n\n● ● ● ● ● ●\n\n● ● ● ●\n● ● ● ● ● ●\n\n● ● ● ● ● ● ● ● ● ●\n\n2 4 6 8 10\n\n-9\n\n-8\n\n-7\n\n-6\n\n-5\n\n-4\n\n-3\n\nn\n\nE\nn\n-\nE\nn0\n,\nm\ngR\ne(\na)\n\n(1\n0\n-\n4\nϵ g\n)\n\nFigure 2 – Energy shift En−E0\nn for antihydrogen interacting with a perfect mirror (blue, top lines), a silicon bulk\n\n(green, bottom lines) or a silica bulk (red, middle lines), in units of 10−4εg. The constant shift corresponding to\nthe real part of mga is represented by the horizontal lines.\n\nThe round trip factor ρ can be approximated as the product of two factors, the quantum\nreflection amplitude r on the CP tail, and a propagation phase factor deduced from the phase\nθ of the Airy functions\n\nρ ' −re2iθ(−zt) , tan θ(x) =\nAi(x)\n\nBi(x)\n, e2iθ(x) = −Ci−(x)\n\nCi+(x)\n. (6)\n\nThe resonance energies En are such that the whole phase on a round trip is an integer multiple\nof 2π (2θ (−zt) + arg (−r) = 2nπ). We then get an effective-range approximation for these\nenergies, which is much more accurate than the scattering length approximation 11.\n\nThe reflection coefficient is a function of the wavevector k and a complex length A(k)\n\nr = −1− ikA(k)\n\n1 + ikA(k)\n, ~k ≡\n\n√\n2mE , (7)\n\nwith the limit A(0) being the scattering length a, while A(k) is now a function of k. For the\nmodel potential exactly described by the homogeneous form V4 ≡ −C4/z\n\n4, the function kA is a\nknown universal function of the dimensionless parameter k`. Its expansion at low values of k is\n\nkA = −ik` α (k`) , ` =\n\n√\n2mC4\n\n~\n, α(K) = α0 + i\n\nπ\n\n3\nK +\n\n(\nα2 +\n\n4\n\n3\nα0 lnK\n\n)\nK2 , (8)\n\nwith known coefficients α0 = 1 and α2 = 8\n3(γ + ln 2)− 28\n\n9 − i2π3 (γ is the Euler constant).\nThe exact potentials describing CP interaction of an antihydrogen atom with perfectly re-\n\nflecting surfaces, silicon or silica bulk contain long-range tails V4 but are not reducible to these\n\n\n\ntails. Using the numerical values obtained for r(k) in 13, we deduce kA by inverting (7) and fit\ncoefficients α0 and α2 in (8) to match its low energy expansion. We then solve the resonance\nenergies as solutions of the equation given after eq.(6).\n\n●\n\n●\n●\n\n●\n\n●\n●\n\n●\n\n●\n\n●\n\n●\n\n●\n\n●\n\n●\n●\n\n●\n●\n\n●\n●\n\n●\n\n●\n\n● ●\n●\n\n●\n\n●\n●\n\n● ●\n\n●\n●\n\n2 4 6 8 10\n\n2\n\n4\n\n6\n\n8\n\nn\n\nE\nn\n-\nR\ne(\nℰ\nn1\n)\n(1\n0\n-\n5\nϵ g\n)\n\n●\n\n●\n\n●\n\n●\n\n●\n\n●\n\n●\n\n●\n\n●\n\n●\n\n●\n\n● ●\n\n●\n\n●\n\n●\n\n●\n●\n\n●\n\n●\n\n●\n\n●\n\n●\n\n●\n\n● ●\n\n●\n\n●\n\n●\n\n●\n\n2 4 6 8 10\n\n-4\n\n-2\n\n0\n\n2\n\n4\n\n6\n\nn\n\nΔ\nE\nn\n(1\n0\n-\n6\nϵ g\n)\n\nFigure 3 – Left plot : Variation of En − λnεg −mgRe(a) for antihydrogen interacting with a perfect mirror (blue\ntop lines), a silicon bulk (green middle lines) or a silica bulk (red bottom lines), in units of 10−5εg. Points are\nobtained from numerical results and full lines interpolate between points. Dashed curves are analytical solutions\nin the effective-range approximation. Right plot : Difference ∆En between analytical and numerical energies\n(same color and line codes as on the left plot; units of 10−6εg.\n\nIn order to assess the precision of the results, we have drawn on the left plot of Figure 3 the\nvariation of En−Re E1n, for the first quantum states of antihydrogen. Points are obtained from the\nnumerical results Enum\n\nn discussed above for n = 1, 2, . . . , 10 with full lines interpolating between\nthese points. Dashed lines are obtained from the solutions Eana\n\nn of the analytical effective-range\napproximation (6-7).\n\nFor completeness, the differences ∆En = Eana\nn −Enum\n\nn between the analytical and numerical\nvalues are also plotted on the right plot of Figure 3. Figure 3 shows small oscillations of the\nnumerical values around the smoother variation obtained from the analytical approximation.\nThese oscillations remain smaller than a few 10−6εg for the first ten quantum states, which\nmeans that the effective-range approximation is sufficient to compute the corrections caused by\nthe CP interaction at this accuracy level.\n\nConclusion\n\nThese results show that the analytical treatment developed in11 with the effective-range approx-\nimation is sufficient to compute the corrections caused by the CP interaction at an accuracy level\nbetter than 10−5εg for the positions of the resonances. This should be sufficient for analyzing\nspectroscopic tests of the weak equivalence principle with antihydrogen 9,10 up to an accuracy\nof this order.\n\nAcknowledgements\n\nThanks are due to M.-T. Jaekel, V.V. Nesvizhevsky, A. Yu. Voronin for insightful discussions\nand to the GBAR and GRANIT collaborations.\n\nReferences\n\n1. V. V. Nesvizhevsky, H. G. Borner, A. K. Petukhov, et al. Quantum states of neutrons in\nthe Earth’s gravitational field. Nature, 415:297–299, 2002.\n\n2. V. V. Nesvizhevsky and A. Yu. Voronin. Surprising quantum bounces. Imperial College\nPress, 2015.\n\n\n\n3. F. Shimizu. Specular reflection of very slow metastable neon atoms from a solid surface.\nPhys. Rev. Letters, 86:987–990, 2001.\n\n4. T. A. Pasquini, Y. Shin, C. Sanner, et al. Quantum reflection from a solid surface at\nnormal incidence. Phys. Rev. Lett., 93:223201, 2004.\n\n5. T. A. Pasquini, M. Saba, G.-B. Jo, et al. Low velocity quantum reflection of Bose-Einstein\ncondensates. Phys. Rev. Letters, 97:093201, 2006.\n\n6. G. Dufour, A. Gerardin, R. Guerout, et al. Quantum reflection of antihydrogen from the\nCasimir potential above matter slabs. Phys. Rev. A, 87:012901, 2013.\n\n7. J. Madronero and H. Friedrich. Influence of realistic atom wall potentials in quantum\nreflection traps. Phys. Rev. A, 75:022902, 2007.\n\n8. P. Indelicato, G. Chardin, P. Grandemange, et al. The GBAR project, or how does\nantimatter fall? Hyperfine Interactions, 228:141–150, 2014.\n\n9. A. Yu Voronin, P. Froelich, and V. V. Nesvizhevsky. Gravitational quantum states of\nantihydrogen. Phys. Rev. A, 83:032903, 2011.\n\n10. A. Yu Voronin, V. V. Nesvizhevsky, G. Dufour, and S. Reynaud. Quantum ballistic\nexperiment on antihydrogen fall. J. Phys. B, 49:054001, 2016.\n\n11. P. P. Crepin, G. Dufour, R. Guerout, A. Lambrecht, and S. Reynaud. Casimir-Polder\nshifts on quantum levitation states. Phys. Rev. A, 95:032501, 2017.\n\n12. G. Dufour, R. Guerout, A. Lambrecht, and S. Reynaud. Quantum reflection and Liouville\ntransformations from wells to walls. EPL, 110:30007, 2015.\n\n13. G. Dufour, R. Guerout, A. Lambrecht, and S. Reynaud. Liouville transformations and\nquantum reflection. J. Phys. 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We calculate the Casimir-Polder shifts of the\nenergies of these states with a precision sufficient to discuss spectroscopy\nexperiments aiming at tests of the weak equivalence principle on antihydrogen.\nGravitationally bound quantum states have been observed with ultracold neutrons\n1,2 , but not with neutral atoms. At the low energies corresponding to such\nbound states, an efficient quantum reflection is produced by the Casimir-Polder\n(CP) tail above a surface 3,4,5,6 , so that quantum levitation states of atoms\ncan be created, where quantum reflection above a horizontal mirror balances\ngravity 7. It has recently be proposed to test the weak equivalence principle\nwith antihydrogen by timing its free fall from a trapping device 8. The\nprecision of this test, of the order of 1%, could be improved by spectroscopic\nmeasurements of quantum levitation states of antihydrogen above a material\nsurface, since the properties of these states are essentially determined by\ngravity 9,10. However an accurate determination of the acceleration requires a\nprecise evaluation of Casimir-Polder shifts on quantum levitation states 11. We\nconsider an atom of mass m and energy E bouncing in the gravity potential mgz\nand reflected on a plane horizontal mirror due to quantum reflection on the CP\npotential V CP (z). The wavefunction ψ(z) obeys the one-dimensional Schrödinger\nequation − 2 2m d 2 ψ dz 2 (z) + V (z)ψ(z) = Eψ(z) , V (z) = mgz + V CP (z) ,\n(1) with the potential V depending only on the altitude z of the particle above\nthe mirror sitting at z = 0. We also suppose that atoms are absorbed when\ntouching the surface, which corresponds to the physical boundary condition for\nantihydrogen annihilated at contact with matter. Length and energy scales\nassociated with the Casimir-Polder potential are respectively much smaller and\nmuch larger than those associated with the gravitational potential g = 2 2m 2 g\n1/3 ≈ 5.87 µm , g = mg g ≈ 0.602 peV , (for g ≈ 9.81 m.s −2). (2) An approximate\nsolution of the problem is thus found by decoupling the effects of gravity and\nCP interactions. For quantum levitation states with low values of the quantum\nnumber n, the scattering amplitudes are mainly given by the scattering length a.\nIt follows that the energies E 0 n","abstract_source_suggest":{"input":"Grobid"}}],"titles":[{"title":"Casimir-Polder shifts on quantum levitation states"}],"external_system_identifiers":[{"schema":"HAL","value":"hal-01802015"}],"facet_author_name":["2152325_Gabriel Dufour","1921399_Romain Gu´erout","2185728_P.P. 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The horizontal axis is the position along the trap axis. The left vertical axis is the voltage seen by particles. The magnetic field strength is shown by the green curve, with the scale on the right vertical axis. In blue, the electron potential well filled with electrons is drawn , reducing the apparent voltage shown by black curve with the value on the left vertical axis. In red is shown the positron potential well. When a positron bunch arrives, the entrance electrode voltage is low (dashed dotted line). It is then increased and positrons go back and forth (it is depicted by red arrows) between this gate and the downstream part of the trap. They pass many times through the electron plasma, and are eventually slowed down and fall into their well. The presence of residual $\\mathrm{H} _{\\mathrm{2}} ^{\\mathrm{+}}$ helps the final catching of positrons.","label":"fig:trapping","source":"arxiv","key":"74e65bec1ab53f8e2d2e3e5ee80e32c2","url":"https://inspirehep.net/files/74e65bec1ab53f8e2d2e3e5ee80e32c2"},{"filename":"acc.png","material":"preprint","caption":"The accumulation of positrons (from \\cite{Grandemange2013}).","label":"fig:acc","source":"arxiv","key":"619c6a23b1adce115bcd77ab329ce342","url":"https://inspirehep.net/files/619c6a23b1adce115bcd77ab329ce342"},{"filename":"potential.png","material":"preprint","caption":"Casimir-Polder (CP) potential for antihydrogen in the vicinity of a material bulk; from top to bottom, perfect conductor (full line), silicon (dashed line), silica (dotted line); (inset:ratio $V/V^*$ to the retarded potential $V^*$ for a perfectly conducting mirror, see text).","label":"fig:CPpotential","source":"arxiv","key":"31a3ff2739594efba66243babcb16c98","url":"https://inspirehep.net/files/31a3ff2739594efba66243babcb16c98"},{"filename":"reflectivity.png","material":"preprint","caption":"Quantum reflection probability $|r|^2$ as a function of the energy for antihydrogen atoms on bulk mirrors; from bottom to top, perfect conductor (full line), silicon (dashed line), silica (dotted line).","label":"fig:reflectivity","source":"arxiv","key":"9f60dd99f652c28dcb40f99e9c6cb3d9","url":"https://inspirehep.net/files/9f60dd99f652c28dcb40f99e9c6cb3d9"},{"filename":"badlands.png","material":"preprint","caption":"The badlands function $Q(x)$ for an antihydrogen atom with energy $E=10$~neV; from bottom to top, perfect conductor (full line), silicon (dashed line), silica (dotted line).","label":"fig:badlands","source":"arxiv","key":"f44f0b6552023bd9faa8d3e9b312e564","url":"https://inspirehep.net/files/f44f0b6552023bd9faa8d3e9b312e564"},{"filename":"shaper.png","material":"preprint","caption":"Scheme of the proposed device to reduce the vertical velocity spread of the falling wavepacket (see text).","label":"fig:shaper","source":"arxiv","key":"4cdded93bc8f85c70236b016f8008173","url":"https://inspirehep.net/files/4cdded93bc8f85c70236b016f8008173"},{"filename":"sketchRes3.png","material":"preprint","caption":"A sketch of the principle scheme of an experiment on magnetically induced resonant transitions between $\\Hb$ gravitational states. 1 - a source of ultracold antihydrogen, 2 - a mirror, 3 - an absorber, 4 - a magnetic field, 5 - a detector.","label":"FigSketch","source":"arxiv","key":"0348144d7b805bc1d42e191bee7c25c5","url":"https://inspirehep.net/files/0348144d7b805bc1d42e191bee7c25c5"},{"filename":"ResLine1.png","material":"preprint","caption":"The transition probability as a function of the magnetic field frequency for the transition from the ground state to $6$-th gravitational state.","label":"FigTrans","source":"arxiv","key":"f3166773a2e3e3181b2d5f75d289a16e","url":"https://inspirehep.net/files/f3166773a2e3e3181b2d5f75d289a16e"},{"filename":"PsLifetime.png","material":"preprint","caption":"Radiative lifetimes of various Ps states as a function of the principal quantum number $n$. The lifetimes were calculated by summing the Einstein $A$ coefficients of all electric-dipole-allowed decay channels from each Rydberg state. For each $A$ coefficient the appropriate radial integrals were determined using analytic expressions for the radial wave functions in a pure Coulomb potential \\cite{Bethe1957}. The dashed line is the annihilation lifetime of $n$s states. After Ref. \\cite{Cassidy2014}.","label":"fig:lifetimesps","source":"arxiv","key":"1c5ab1eb23c1228c910748220dae3f5a","url":"https://inspirehep.net/files/1c5ab1eb23c1228c910748220dae3f5a"},{"filename":"PsScheme.png","material":"preprint","caption":"A schematic representation of a Mills-Leventhal type of Ps free fall experiment. A real experiment will undoubtedly be significantly different from this illustration, which is intended only to highlight some of the different steps involved. Of distinct practical concern will be the need to keep the apparatus at low temperatures to mitigate effects of black body radiation, as well as minimizing the Ps speed, which will determine the length of the flight path, and hence the experiment.","label":"fig:expscheme","source":"arxiv","key":"1df028fe41187b61f159daeb454a0dd8","url":"https://inspirehep.net/files/1df028fe41187b61f159daeb454a0dd8"},{"filename":"StarkShift.png","material":"preprint","caption":"Stark states of $n$ = 30 and 31 states of Ps, with $m$ = 2 (grey dashed) and $m$ = 29 (black). In the $n$ = 30 level the $m$ = 29 state is a circular state and experiences no first-order Stark shift and only a very weak second-order shift, as explained in the text.","label":"fig:starkstates","source":"arxiv","key":"b57d7747b352a9ea40e1468fa569d53e","url":"https://inspirehep.net/files/b57d7747b352a9ea40e1468fa569d53e"},{"filename":"SchemeQMBounce.png","material":"preprint","caption":"Possible scheme for the observation of the gravitational quantum states of positronium.","label":"fig:SchemeQMBounce","source":"arxiv","key":"96ef6501dc660c224a149f4a9858acec","url":"https://inspirehep.net/files/96ef6501dc660c224a149f4a9858acec"}],"legacy_version":"20190805230903.0","inspire_categories":[{"term":"Gravitation and Cosmology"},{"term":"General Physics"},{"term":"Quantum Physics","source":"arxiv"}],"accelerator_experiments":[{"legacy_name":"CERN-GBAR","record":{"$ref":"https://inspirehep.net/api/experiments/1108191"}}],"first_author":{"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/04ex24z53"},{"schema":"ROR","value":"https://ror.org/01h14ww21"}],"full_name":"Dufour, Gabriel","last_name":"Dufour","first_name":"Gabriel","recid":2152325},"control_number":1313587,"dois":[{"source":"bibmatch","value":"10.1155/2015/379642"},{"material":"publication","source":"arXiv","value":"10.1155/2015/379642."}],"document_type":["conference paper"],"texkeys":["Dufour:2014pea"],"abstracts":[{"source":"HINDAWI","value":"Different experiments are ongoing to measure the effect of gravity on cold neutral antimatter atoms such as positronium, muonium, and antihydrogen. Among those, the project GBAR at CERN aims to measure precisely the gravitational fall of ultracold antihydrogen atoms. In the ultracold regime, the interaction of antihydrogen atoms with a surface is governed by the phenomenon of quantum reflection which results in bouncing of antihydrogen atoms on matter surfaces. This allows the application of a filtering scheme to increase the precision of the free fall measurement. In the ultimate limit of smallest vertical velocities, antihydrogen atoms are settled in gravitational quantum states in close analogy to ultracold neutrons (UCNs). Positronium is another neutral system involving antimatter for which free fall under gravity is currently being investigated at UCL. Building on the experimental techniques under development for the free fall measurement, gravitational quantum states could also be observed in positronium. In this contribution, we report on the status of the ongoing experiments and discuss the prospects of observing gravitational quantum states of antimatter and their implications.","abstract_source_suggest":{"input":"HINDAWI"}},{"source":"arXiv","value":"Different experiments are ongoing to measure the effect of gravity on cold neutral antimatter atoms such as positronium, muonium and antihydrogen. Among those, the project GBAR in CERN aims to measure precisely the gravitational fall of ultracold antihydrogen atoms. In the ultracold regime, the interaction of antihydrogen atoms with a surface is governed by the phenomenon of quantum reflection which results in bouncing of antihydrogen atoms on matter surfaces. This allows the application of a filtering scheme to increase the precision of the free fall measurement. In the ultimate limit of smallest vertical velocities, antihydrogen atoms are settled in gravitational quantum states in close analogy to ultracold neutrons (UCNs). Positronium is another neutral system involving antimatter for which free fall under gravity is currently being investigated at UCL. Building on the experimental techniques under development for the free fall measurement, gravitational quantum states could also be observed in positronium. In this contribution, we review the status of the ongoing experiments and discuss the prospects of observing gravitational quantum states of antimatter and their implications.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["gr-qc"],"titles":[{"source":"arXiv","title":"Prospects for studies of the free fall and gravitational quantum states of antimatter"}],"imprints":[{"date":"2015"}],"curated":true},"id":"1313587","created":"2014-09-03T00:00:00+00:00"},{"updated":"2025-10-01T12:26:05.147592+00:00","links":{"bibtex":"https://inspirehep.net/api/literature/1476804?format=bibtex","latex-eu":"https://inspirehep.net/api/literature/1476804?format=latex-eu","latex-us":"https://inspirehep.net/api/literature/1476804?format=latex-us","json":"https://inspirehep.net/api/literature/1476804?format=json","json-expanded":"https://inspirehep.net/api/literature/1476804?format=json-expanded","cv":"https://inspirehep.net/api/literature/1476804?format=cv","citations":"https://inspirehep.net/api/literature/?q=refersto%3Arecid%3A1476804"},"metadata":{"publication_info":[{"journal_volume":"101","artid":"160403","year":2008,"journal_record":{"$ref":"https://inspirehep.net/api/journals/1214495"},"journal_issue":"16","journal_title":"Phys.Rev.Lett."}],"authors":[{"raw_affiliations":[{"value":"Laboratoire Kastler Brossel, CNRS, ENS, UPMC, Campus Jussieu case 74, 75252 Paris, France"}],"affiliations_identifiers":[{"schema":"ROR","value":"https://ror.org/01h14ww21"}],"full_name_unicode_normalized":"lambrecht, astrid","full_name":"Lambrecht, Astrid","record":{"$ref":"https://inspirehep.net/api/authors/1001198"},"last_name":"Lambrecht","ids":[{"schema":"INSPIRE BAI","value":"A.Lambrecht.1"}],"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/907500"},"value":"Paris, Lab. 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