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v9_16-01-06.doc - HAL

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    A TABULATION AND CRITICAL ANALYSIS OF

    THE WAVELENGTH-DEPENDENT

    DIELECTRIC IMAGE COEFFICIENT

    FOR THE INTERACTION EXERTED

    BY A SURFACE

    ONTO A NEIGHBOURING EXCITED ATOM

    (1,2)(1)*(1)Solomon Saltiel , Daniel Bloch and Martial Ducloy

     (1) Laboratoire de Physique des Lasers, UMR7538 du CNRS et de l'Université Paris13.

    99, Av. J.B. Clément, F-93430 Villetaneuse, France

     (2) Physics Department, Sofia University, 5, J. Bourchier Blvd., 1164 Sofia, Bulgaria

    * e-mail: bloch@lpl.univ-paris13.fr

Abstract

    The near-field interaction of an atom with a dielectric surface is inversely proportional to the cube to the distance to the surface, and its coupling strength depends on a dielectric image coefficient. This coefficient, simply given in a pure electrostatic approach by (-1) / (+1) with the permittivity, is specific to the frequency of each of the various relevant atomic transition : it depends in a complex manner from the bulk material properties, and can exhibit resonances connected to the surface polariton modes. We list here the surface resonances for about a hundred of optical windows whose bulk properties are currently tabulated. The study concentrates on the infrared domain because it is the most relevant for atom-surface interaction. Aside from this tabulation, we discuss simple hints to estimate the position of surface resonances, and how uncertainties in the bulk data for the material dramatically affect the predictions for the image coefficient. We also evaluate the contribution of UV resonances of the material to the non resonant part of the image coefficient.

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1. Introduction

    Atomic Physics and the related high-resolution sensitive spectroscopy techniques allow for the probing of long-range atom-surface interaction [1] with a high accuracy. Recently, it has been experimentally demonstrated that the universal van der Waals (vW)

    -3attraction between an atom and a neighbouring surface, that spans in z with z the atom-

    surface distance, could be turned into a repulsion [2,3] through a resonant coupling between virtual atomic transitions and resonances of the surface. It was also shown [4] that in a related process, an excited atom can undergo a remote quenching to a lower energy state analogous to a Förster-type energy transfer here applied to the surface mode. The long-range coupling to the surface can indeed open an energy-transfer channel, that would remain otherwise nearly prohibited for spontaneous emission in the vacuum. More generally, the development of various techniques confining cold atoms close to surfaces and the attempts to selectively deposit atoms or thin layers for nanofabrication purposes, induce a growing need for the control and engineering of the atom-surface interaction.

    It is the purpose of this paper to provide in a simple manner, and for a large set of materials, the surface-related parameters determining the atom-surface interaction. Because the atom-surface interaction can be expanded over the various atomic transitions to coupled levels, the specific properties of the considered dense material can be determined by a simple image coefficient” (relative to an ideal reflecting surface), defined for each relevant atomic coupling. As recalled below, these coefficients are in the principle deduced from the spectral knowledge of the bulk permittivity of the material (?), through a complex (planar) surface

    response function S, that simply turns to be S() = ( - 1) / ( + 1) for a non-dispersive

    material.

    The paper is presented in the following way. In section 2, we briefly recall the essential results for the physics of the atom-surface interaction in the near-field regime, in order to provide in an intelligible manner the reflection coefficients applicable for a virtual transition in absorption, as well as for a virtual emission, and the dielectric coefficient

    relevant for a real energy transfer. Emphasis is on these atomic emission processes occurring

    only for excited atoms-, as they are susceptible to couple resonantly with the surface mode resonances naturally appearing in the surface response function S [5]. Section 3 is mostly devoted to a listing of the surface resonances obtained for a large list of optical materials, essentially those whose bulk values are known from the Palik Handbook [6] tabulation, or for

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    which a fitting expression for (?) is published in the literature. We concentrate on

    resonances in the IR domain, and hence on dielectric and semi-conductor materials, because the IR contributions usually provide the dominant vW surface interaction, and because "optical" materials most often exhibit a transparency window in the visible range (i.e. no

    resonance should appear in the visible). Although the presented results are derived from a numerical evaluation, we also discuss a simple method to approximately locate the resonance. Section 4 discusses the issue of accuracy of the predictions for a resonant behaviour, showing that apparently minor discrepancies between published data for the bulk material may lead to dramatically differing predictions for the surface behaviour. This is illustrated with the examples of AlSb, InSb and YAG, and then discussed on a more general basis : in particular, it is shown that the original data -notably reflectivity studies- from which the bulk permittivity is usually extracted, can be more relevant than the use of tabulated or spectrally modelled values of permittivity. Aside from the resonant behaviour, an accurate determination of the nonresonant contributions can also be needed, notably because the effective atomic behaviour usually results from a summing of various contributions, most of them non resonant. This implies that the specific "exotic" behaviour (e.g. repulsion) induced by a

    resonant term can be strongly corrected by the additional non resonant terms : section 5 concentrates on the smoothly frequency-varying non resonant contribution from tabulated bulk values. It is in particular shown that the UV resonances, although effective only through their far wings response, are far from being entirely negligible.

    2. Atom interaction with a dielectric medium in the range of near-field electrostatic approximation

    2.A Energy shift and virtual transitions

     In the vicinity of a perfect reflector, an atom in the level?i> undergoes a dipole-dipole

    interaction expressed as :

    221ijij??;;Vz~~ (1) ???iz3??16zjijwith ~ the dipole moments related to the virtual transition ?i> ( ?j> . In eq (1), we assume

    that the retardation effects are negligible in order to ensure the electrostatic approximation, i.e.

    z << , with the wavelength of the?i> ( ?j> transition; also, we typically assume z ) 1 ijij

    nm, in order to be insensitive to the structural details of the surface. Such a description, with

    -3its z spatial dependence, is known to characterize the non-retarded atom-surface vW interaction, often described as the dipole coupling between a (fluctuating) atom dipole and its

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    (instantaneously correlated) image induced in the reflecting surface. Note that through the summing over the dipole couplings appearing in eq.(1), the influence of IR transitions between atomic levels of neighbouring energy is strongly enhanced relatively to their relative weight in a spontaneous emission process (see [1]). This is why in the following, our focusing will be on the IR resonances of materials.

    If the neighboring surface is not a perfect reflector, but a dielectric medium, the energy shift has to be modified in the following way:

    221ijij??Vz r(?) (2) ;;~~???ijiz3??n16z

with r(?) a "dielectric image coefficient" affecting the virtual transition ?i>(?j>. If the ij

    dispersion of the dielectric medium could be neglected (i.e. the dielectric permittivity is

    constant over the whole spectrum), this dielectric coefficient would be frequency-independent and simply given by the electrostatic image coefficient r = ( - 1) / ( + 1). More correctly,

    when the dispersive features of the dielectric coefficient are taken into account [7], one finds for a virtual absorption (i.e. ?> 0) : ij

    ?ω2ε(iu)1ij (3) r(ω)duaij22?0πωuε(iu)1ij

    and for a virtual emission (i.e. ?< 0) : ij

    ??ε(ω)1ω2ε(iu)1ij?ijr(ω)du2e (4) ???eij022πωuε(iu)1ε(ω)1??ijij??

    Eq. (4) can be also written as :

    r(ω)r(ω)2eS(ω) (5) eijaijij

    where we have introduced in (5) the surface response function S(?) = [(?) - 1]/ [(?) + 1].

    From eqs. (3-5) one notes that, once the dispersion of the dielectric permittivity is taken into account, the knowledge of the permittivity on the whole spectrum is required. However, in eq. (3), i.e. for the case of a virtual absorption (?> 0), causality and the ij

    Kramers-Krönig relationship impose the boundaries 0 < r(?) < 1 along with a monotone

    behaviour for r(?) as a function of ?. Hence, one understands that the accuracy on r(?)

    depends only smoothly upon the uncertainties in the determination of (?). Conversely, for a

    virtual emission of the atom (?< 0, eqs. (4-5)), there appears a second term in the dielectric ij

    coefficient that is susceptible to evolve arbitrarily: its amplitude can possibly exceed unity, its

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    sign can be positive or negative. These features have been analyzed [7,8] as originating in a resonance between a virtual absorption into a surface-plasmon [7] or a surface-polariton mode [8], and the atom emission. They are strongly dependent upon the spectral features of the dielectric medium.

2.B Surface-modified decay rate

     In addition to the energy-shift induced by the vicinity with the surface, which even affects an atom in its ground-state, the decay rate of an excited atom, and the relative efficiency of the various de-excitation channels, can depend sharply on the vicinity with a surface. For our discussion, centred on the resonant effects, we do not consider the finite increase of the decay rate in the presence of a transparent dielectric surface, related with an enhanced spontaneous emission through the near-field evanescent-wave coupling between the emitting atom and the surface [1,9]. Rather, we consider the case when the bulk material is absorbing at the frequency associated to an IR transition between the excited atomic level and a neighbouring lower energy level. This decay channel -usually in the mid-IR range and hence

    -3 often very weak for an atom in the vacuum- undergoes a strong zmagnification in the

    vicinity with the surface, through a dissipative analogous of the resonant enhancement of the van der Waals interaction [1,4,7]. The atomic decay rate for the ?i> (? j> process varies ij

    as :

    ?????()1ij3????(z)(?)1(2?z/)m (6) ijijij??()1???ij????

    !,;;;;mε(?)1ε(?)1 The notable result of eq.(6) is the appearance of the factor = ijij

    m[S(?)]eS(?), which is the dissipative counterpart of the resonant term involved in ijij

    m[S(?)]eqs. (4-5). This factor governs the distance at which the surface-induced decay ij

    channel becomes predominant relatively to standard spontaneous emission.

3. Surface resonances of materials

    As discussed in section 2, the most "exotic" behaviors induced by a resonant coupling between the atomic excitation and the surface polariton mode, are characterized by the complex surface response S(?) as defined following eq.(5). Conversely, the non resonant ij

    contribution r(???) provides a contribution varying only smoothly with the energy of the aij

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    atomic transition. These terms remain however important in the final summing of all virtual contributions, and cannot be ignored in the final assessment of the surface interaction.

    e[S(?)]As it is well-known, and will be further exemplified in section 4, is

    m[S(?)] absorption-like. A simplified modeling of the essentially dispersion-like, and

    permittivity (?) - notably those extrapolated from a dilute medium approach- would fully justify this point. Such a view is only approximate because resonances in dense media are much broader than current atomic resonances, and because the overlap of several neighbouring resonances most often precludes a perfect (anti-)symmetry. Nevertheless, a bunch of useful information can be described with the position, width, and amplitude of these resonances.

    Figure 1 and Table 1 constitute the core of the paper, and characterize the surface resonances for numerous optical materials. The values of the bulk permittivity (?) are mostly

    taken from the compiled values provided in the Palik handbook [6] or from fitting expressions

    1/2for (?) ; for birefringent materials, the permittivity is obtained by taking () [10], the //(

    value that applies for a symmetry axis oriented towards the normal to the surface, making the cylindrical symmetry not broken in spite of the birefringence. As already discussed at length for the case of sapphire in [8], the surface resonances actually occur for radically differing frequencies than those of the bulk material. For the clarity of presentation, in figure 1, we have defined the position of the resonance(s) of a given material as the frequency associated

    m[S(?)]to the peak value of the nearly absorption-like ; for the amplitude, we characterize

    m[S(?)]m[S(?)]m[S(?)] by its peak value (one has ) 0, and = 0 in the transparency

    e[S(?)]window), and the nearly dispersion-like by its extreme values. Note that these

    m[S(?)]extreme values would be opposite and simply related to the amplitude in the case

    of an ideal narrow and well-isolated resonance. In addition and to further characterize a resonance with an indication of its width, Table 1 provides the frequency positions of the

    e[S(?)]extreme peaks of . Aside from these essential features, the general behavior of these resonances, including their far extended wings, can be calculated by directly applying the tabulated values of the complex index n+i to evaluate S(?) (through = (n+i)?).

    The information provided in table 1 and figure1, should make easy the selection of the right material if a resonance with a specific atomic excitation is needed. Oppositely, it also allows one to predict when the effect of a narrow resonance can be ruled out. In all cases, one

    e[S(?)]has to keep in mind that the dispersive resonance for implies slowly decaying tails,

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    so that approximate coincidences, leading to resonant behaviours, are relatively easy to find. Note also that the resonant nature of the atom-surface van der Waals is truly dominant only

    e[S(?)]when ?? is at least comparable with unity -or with r(?) -. Conversely, at the a

    m[S(?)]smaller distance, even a relatively small value for induces large changes of the

    lifetime and branching ratios : this is because there is no equivalent of a "non resonant" change for this dissipative effect.

    Aside from these numerical evaluations, it is possible to assess an approximate location of the S(?) resonances. Before, it is worth nothing that in the theory, the resonance

    -1is obtained for a pole of [(?) +1], but that this pole is at a complex frequency. Surface

    resonances are usually so broad that the complex pole frequency is not very useful for a practical location of surface resonances, notably the tiny ones.

    As can be seen from Table 1, the "centre" of the resonance, as defined through the

    m[S(?)]peak frequency of , is very close to the centre of the anomalous dispersion (for e[S(?)]), and in most cases (for pronounced resonances) close to the zero value of e[S(?)] (see figure 1). With the complex permittivity provided through the complex index

    (n+i), one gets :

    (?)12(n?? 14inS(?)1 (7) (?)1(n??1)2in(n??1) 4n??

    e[S(?)]so that the "resonance" (when defined by =0) occurs for :

    n?+? = 1 (8)

    With this relation, one easily shows that = /n, and surface resonances will m[S(?)]res

    appear only if n is small enough, and thus ( =) close to unity. If n<<1, the 1n?

    resonance amplitude is on the order of 1/n. It is also worth noting that the so-defined resonance condition can be read as?;?=1, a condition that is satisfied by the pole condition

    (for complex frequency) = -1.

     The interest for such a simple estimate is twofold : on the one hand, it provides, in a very elementary manner, a way to locate and characterize a surface resonance from the knowledge of optical values characterizing the bulk material; moreover, this estimate does not depend of a specific modeling of the bulk resonance. On the other hand, it shows that these surface resonances always occur in a frequency region where the optical material is strongly absorbing (typically on half a reduced wavelength), so that the material is no longer an optical "window", implying specific difficulties in the evaluation of its optical constants. In the next

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    section, we discuss the issue of the uncertainties in the tabulated data, with respect to the fact

    e[S(?)]that the exact features of the surface response S(?) - and notably the sign of , upon

    which is based the prediction of a vW attraction or repulsion- are strongly dependent on the accuracy of the determination of n and .

4. Selecting bulk data to evaluate the surface resonance

     It is naturally not an uncommon situation that measurements performed by various authors for the same material lead to accidental differences in the tabulated optical constants. The use of different samples, or differing experimental conditions, such as the temperature of the sample, may unsurprisingly lead to some discrepancies. More fundamentally, the spectral determination of a pair of optical constants (n,) that are experimentally intricate, usually

    demands an amount of extrapolation. When the evaluation relies on the Kramers-Kronig relationship, the knowledge of the whole spectrum is even requested. However, when the goal of these optical analyses on the bulk material is to determine the volume resonances of a material, the final discrepancies usually appear to be relatively minor and insensitive to the absolute calibration of the optical measurements. Conversely, these marginal uncertainties lead to dramatic changes for surface resonances.

    We illustrate below such situations. As a first example, we consider the case of AlSb, that features a single resonance in the far IR, and for which two sets of data for (n,) are

    provided in [6], based upon two different experimental studies [14,15]. In figure 2a, the comparison of the plotted values for n and according to the two different sets of data

    exhibits notable differences in some values, but no major discrepancies in the position of the peaks for these bulk parameters. However, the frequency where ~1 is strongly dependent on

    the choice of data. This explains that, as shown by fig.2b, the location of the predicted surface resonances is critically dependent on the considered set of (n,) values. Conversely, the

    resonant behaviour of S(?) in the wings of the surface resonance appears independent of the quality of the bulk data. Also, an analytical modeling of the bulk resonances (e.g. classical

    theory of dispersion), involving a limited number of parameters can be considered [14,15]: it usually leads to slightly modified values of the (n, ) set and to slightly sharper surface

    resonances, but does not essentialy alter the position of the surface resonances as deduced from an extrapolation of the tabulated values in [6]. A second illustration is provided by InSb, with two bulk resonances in the far IR : although the discrepancies occurring between the two sets of data (Fig.3a) are comparable for both resonances, one notices (fig. 3b) that one of the

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    -1surface resonances (the one with the lower energy around 70 cm, effectively measured by a

    Fourier transform method in [40], otherwise only extrapolated from IR data in [41]) is much

    -1 more sensitive than the other one (around 190 cm)to the choice of the set of bulk parameters.

    As an additional example, in the less remote IR range, YAG is a genuine dielectric (non semi-conductor) medium of a great practical importance (including for our own experiments with the vW interaction, see [3]) : it exhibits multiple bulk resonances, partly shown in fig. 4a. As for InSb, some of the surface resonances (fig. 4b) are extremely sensitive to the exact assessment of the bulk resonances. In addition to these simple illustrative examples, similar remarks could be derived from the differing sets of (n,) values found for example for GaAs,

    or for BaF, although some critical considerations may help to choose among the data 2

    proposed in the literature (see table 1).

    As already mentioned, the set of (n,) value is usually not directly measured, and

    requires a disentanglement to be obtained. Among the current techniques to get these (n,)

    values, the measurement of reflectivity close to the normal incidence appears to be particularly relevant for these issues of surface resonances. It is possible to reconstruct the reflectivity from the (n,) data, given either by discrete tabulated values, or by an analytical modelling. As shown in fig 2c, 3c, 4c, a correlation appears between the sensitivity of the reflection spectrum to the considered set of data, and the predictions for the surface resonance. In most cases, the strongest disagreement between various sets of data is not for the position of the peaks of reflectivity, but rather occurs in the sharp wings of the reflectivity spectrum : there can be some discrepancies in the absolute values of reflectivities around the peaks, or in the typical "width" of the reflectivity resonance, but the most radical variations appear in the reflectivity values around these wings when comparing various sets of data. This connection between reflectivity and the surface response, can be understood from the Fresnel formulae for normal incidence. The reflectivity (in intensity) R(?) being given by:

    2ni1(n1) ?4nR(?)1 (9) ni1(n1) ?(n1) ?

    one sees that R(?) ~1 in the regions of strong bulk absorption (characterized by >>1), while

    close to a surface resonance- eq.(8), one has rather R(?) ~ (1- n)/(1+n). If sharp surface

    resonances are characterized by ~ 1, and n << 1, however, most of the surface resonances,

    when not an extremely sharp one, rather occur for ? 1 and an arbitrary value of n (n?1). In

    some cases, the experimental data directly measure the reflectivity, with uncertainties mostly originating from the absolute reflectivity calibration (e.g. for non evacuated systems, at

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    wavelengths known for air absorption), or possibly from the wavelength selection system (especially for older apparatus), or from the imperfections of the surface state, responsible for a possible light scattering (although scattering losses are expected to be small in the IR range). These remarks show that when the literature is not precise enough to provide a reliable value of the resonant behaviour at a given wavelength, it should be sufficient to measure around the wavelength of interest the reflectivity of the window, in conditions (e.g. temperature) similar

    as close as possible as those used for the planned experiments. In this spirit, we had performed reflectivity measurements of two YAG windows on vapour cells currently used for our studies (fig. 4). They tend to establish that the data of ref [37] (used for our predictions in

    -1 [3] for the ~ 820 cmresonance), is most probably irrelevant, at least for the YAG samples that we use.

5. The non resonant contribution r(?) and the influence of the UV absorption a

     As recalled in section 3, the non resonant contribution r(?) exhibits a smooth a

    monotone decrease with ?. Its intrinsic integration of fluctuation properties over the whole spectrum makes it remarkably insensitive to the uncertainties affecting the bulk properties. However, the evaluation of the precise behaviour of an atom - in a given state- in front of a surface, with its summing over numerous coupling transitions, may demand some accuracy in the evaluation of the r(?) values. We discuss here some of the possible approaches for the a

    evaluation of r(?). a

     Because of the relative insensitivity of r(?) to the details of the bulk permittivity, and a

    because of the imaginary frequency appearing in eq.(3), it is very convenient to use, when available, an analytical expression for (?), enabling an easy extension and calculation in the

    complex plane. However, in most cases (one of the few exceptions is for sapphire, see [13]), these analytical expressions are limited to the band of IR absorption band, and are irrelevant inside the transparency window, or in the UV absorption band. In the absence of an experimentally determined analytical expression spanning over the whole spectrum, r(?) is a

    numerically evaluated from its real-valued equivalent expression [8]:

    ??20 (10) r(?)Pm[S(?)]d?e[S(?)]?a00?0????0

    where P stands for the Cauchy principal value. Actually, when an analytical formula for (?)

    can be found for the IR part of the spectrum extending up to the large transparency window in the "visible" range, an approach combining the analytical integration for the IR range, and the

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