Luc Deike

Claude Laroche

Eric Falcon

Experimental study of the inverse cascade in gravity wave turbulence

Keywords: 47, 35, -i -Hydrodynamic waves 05, 45, -a -Nonlinear dynamics and chaos 47, 27, -i -Turbulent flows

Introduction. -Waves on ocean surface is the most common example of wave turbulence. Wave turbulence concerns the dynamical and statistical study of a field of dispersive waves in nonlinear interaction. It is a rather universal phenomenon since it occurs in various physical contexts including geophysics, plasma physics, non linear optics or solid state physics [START_REF] Nazarenko | Wave Turbulence[END_REF]. One of the most important result of wave turbulence theory is the existence of out-of-equilibrium stationary solutions for the wave spectrum that follow Kolmogorov-like cascades of flux of conserved quantities [START_REF] Nazarenko | Wave Turbulence[END_REF]. This cascade type behavior is similar to two-dimensional turbulence ones where both a direct cascade of enstrophy (rms vorticity) and an inverse cascade of energy occur [START_REF] Kraichnan | [END_REF]3]. In wave turbulence, the cascades are governed by the nonlinear interaction process between waves: for a 4-wave process (as for gravity surface waves) the energy and the wave action are conserved [START_REF] Nazarenko | Wave Turbulence[END_REF]4], whereas for a 3-wave process (as for capillary waves) only the energy is conserved [START_REF] Nazarenko | Wave Turbulence[END_REF]. Frequency-power law solutions for gravity wave spectrum then exist corresponding to either a constant flux of energy from large to small scale (direct cascade) or a constant flux of wave action from small to large scale (inverse cascade) [5]. The direct gravity cascade has been observed in open seas [6], in wellcontrolled laboratory experiment [7,8], and in numerical (a) Corresponding author: luc.deike@univ-paris-diderot.fr simulations [9]. Existence of an inverse cascade in gravity wave turbulence has been confirmed recently using numerical simulations [10,11]. These simulations notably show the formation of both inverse and direct gravity cascades without interacting each other and of the "condensation" of waves at large scale as an analogous of Bose-Einstein condensation in condensed matter physics [11]. Experiments related to the observation of an inverse cascade in wave turbulence are scarce. It concerns either non linear optics [12] or heat wave turbulence in a superfluid [13]. To our knowledge, no observation of an inverse cascade in gravity wave turbulence has been reported so far.

In this letter, we report the observation of upscale transfers in gravity wave turbulence. The energy injected into the waves at an intermediate scale (corresponding to the gravity-capillary length) generates waves of larger and larger scales due to the nonlinear wave interactions. A stationary state is reached and presents properties close to the inverse cascade of wave action predicted by wave turbulence theory [START_REF] Nazarenko | Wave Turbulence[END_REF]. We show that the large scale cut-off of this upscale transfer regime is related to the horizontal finite size of the set-up. We characterized this regime by measuring the scaling of the wave amplitude spectrum with the frequency scale and with the power injected into the waves. The probability density of the injected power fluctuations is also measured in the presence or in the absence of the upscale transfer regime of gravity wave tur-bulence.

Experimental set-up. -The experimental setup is similar to the one used in [7]. It consists in a square plastic vessel, L = 20 cm side, filled with mercury up to a height h = 18 mm. The properties of mercury are: density, ρ = 13.5 × 10 3 kg/m 3 , kinematic viscosity, ν = 10 -7 m 2 /s and surface tension γ = 0.4 N/m. Mercury is used because of its low kinematic viscosity. Surface waves are generated by a rectangular plunging wave maker (13 cm in length and 3.5 cm in height) driven by an electromagnetic vibration exciter. The crossover frequency between gravity and capillary linear waves is f gc = 1 2π 2g/l c ≃ 17 Hz with g = 9.81 m/s 2 the acceleration of the gravity, and l c = γ/(ρg) is the capillary length [7]. Gravity waves thus occurs for frequency f < f gc whereas capillary waves occurs for f > f gc . The wave maker is driven around f gc in order to generate small scale gravity waves to be able to observe an upscale transfer regime from this small scale to larger ones. The wave maker is either driven sinusoidally at a frequency f p = 19 Hz close to f gc or with a random noise (in amplitude and frequency) band-pass filtered around f p ± 3 Hz unless otherwise stated. The depth H of the wave maker immersion is varied in a range 9 ≤ H ≤ 17 mm. The amplitude of the surface waves η(t) at a given location is measured by a capacitive wire gauge plunging perpendicularly to the fluid at rest [7]. The frequency cut-off of this probe is near 400 Hz. The signal η(t) is recorded during 500 s using an acquisition card with a 2 kHz sampling rate. The instantaneous injected power into the fluid I(t) is given by the product of the wave maker velocity V (t) and the force F (t) applied by the vibration exciter to the wave maker [7]. The mean injected power is thus I ≡ F (t)V (t) where • denotes a time average. σ F and σ V will denote the rms value of F (t) and V (t).

Wave amplitude power spectrum. -Figure 1 shows the power spectrum density of the wave amplitude S η (f ) for different forcing amplitudes when the wave maker is driven sinusoidally at f p ≃ f gc . At low forcing amplitude (bottom curve), the power spectrum exhibits a set of peaks corresponding to the excitation frequency f p and the harmonic ones nf p , with n an integer. We observe also the presence of two peaks of small amplitudes at adjacent side-band frequencies f p ± δω, and a peak at δω. These latter are due to the Benjamin-Feir instability that destabilizes a monochromatic weakly non-linear wave train in deep-water into a wave train modulated in amplitude with a modulation frequency δω [14]. When the forcing amplitude is increased, the growth of the peak amplitudes at the driving and harmonic frequencies is observed as well as a broadening in frequency. For the maximum forcing amplitude (top curve), the peak at f p and its harmonics are still observed whereas the peak amplitude at δω has decreased since the frequency broadening of the fundamental is larger than δω (curves of Fig. 1 are shifted vertically for clarity). Similar results are found when the frequency of the sinusoidal forcing is varied around the crossover fre- quency f gc . Even for the maximum accessible amplitude of this forcing (top curve in Fig. 1), the spectrum is still discrete with no power-law spectrum neither in the low frequency part of the spectrum nor in the high frequency part. An easy way to increase the forcing energy at this crossover scale is to use, instead of a sinusoidal forcing, a random one within a narrow frequency bandwidth around f gc .

Figure 2 shows the power spectrum density of the wave amplitude S η (f ) for different forcing amplitudes when the wave maker is driven with a band-pass filtered random noise in a frequency range f p ∈ [16,22] Hz. At low forcing amplitude (bottom curve), the peak at the forcing frequencies slightly broadens and its harmonic near 2f p appears. No power law is observed neither for f > f p nor for f < f p . When the forcing amplitude is increased, the spectrum amplitude at f p does not increase whereas the ones at frequencies located on both sides of f p grew significantly. For high enough forcing and for f < f p (corresponding to the gravity wave regime), a power law spectrum is observed from the forcing scale (f p ) up to a larger scale over a range less than one decade in frequency (see top curve and dashed line in Fig. 2). Although, quite narrow in frequency, this range corresponds to more than one decade in wave number k using the dispersion relation of linear gravity waves ω(k) = √ gk with ω ≡ 2πf , k ≡ 2π/λ and λ the wavelength. This power-law spectrum is found to scale as f -α with α = 3.3 ± 0.2 and could come from the upscale wave action transfers of gravity wave turbulence. The gravity wave power spectrum for such an inverse cascade is predicted as S theo η ∼ q1/3 gf -11/3 [START_REF] Nazarenko | Wave Turbulence[END_REF]5] where q is the mean wave action flux (q has dimension of [ε]/[ω], i.e. L 3 /T 2 , with ε the mean energy flux). The exper- imental exponent, α = 3.3 ± 0.2, found here is thus in rough agreement with the predicted one 11/3 ≃ 3.6. For f > f p (corresponding to the capillary wave regime), the spectrum has a rounded shape for f f p , and displays a power-law for higher frequencies (see top curve and dotdashed line in Fig. 2). This latter is related to capillary wave turbulence although the power-law exponent found here (-3.6) is larger than the expected one (-17/6) by weak turbulence theory [START_REF] Nazarenko | Wave Turbulence[END_REF]. The rounded shape of the spectrum linking the inverse cascade of gravity waves and the direct cascade of capillary waves could be due to an accumulation of energy flux in this region. Indeed, for gravity waves, an inverse cascade of wave action is theoretically accompanied with a direct cascade of energy flux [START_REF] Nazarenko | Wave Turbulence[END_REF].

Note that the power-law of the inverse regime of gravity waves is typically reached 50 s after the starting of the forcing. It thus not allow us to collect enough statistics to accurately studied the transient regime of the dynamical growth of these upscale transfers.

Discussion. -One can wonder if the above observation of an upscale wave generation could be ascribed to a mechanism different than the inverse cascade one suggested here.

i) Cross waves generated by a wave maker by parametric instability are known to propagate at a frequency half of the driving one [15], and could thus polluted the wave spectrum at small scales. However, the onset of occurrence of cross-waves theoretically depends on the amount of water displaced by the wave maker [16]. Thus, in our experiments, the immersed part of the wave maker has been decreased to reduce the amount of water displaced by this latter. When the immersion depth H of the wave maker is lower than two thirds of the fluid depth h (i.e. H < 12 mm), we find that the power spectrum of the wave amplitude does not exhibit a peak at half frequency the driving one in contrast with the one observed for a deep plunging wave maker. No cross wave is thus emitted here when H is small enough (as it is the case in all figures).

ii) For our depth of fluid (h = 18 mm), waves of wavelengths λ ≪ 12 cm are in a deep-water regime (kh ≫ 1). The continuum power law in the gravity regime is observed in Fig. 2 for f > 4 Hz corresponding to wavelengths λ < 8.6 cm (kh > 1.3). One thus probes kh > 1. Depth effects on wave turbulence properties have been observed to occur for smaller depths (typically when h < 12 mm for the same frequency range as here) [17]. The upscale wave generation is thus not related to a depth effect.

iii) The upscale wave generation observed here is not also related to a zero-frequency modulation instability [18] nor to a defect-mediated turbulent regime reported [19]. In these studies, the amplitude of the wave spectrum at zero-frequency is of the order of the maximum amplitude of the spectrum. Here, the amplitude of the zero-frequency spectrum is negligible compared to the maximum amplitude of the spectrum (see Fig. 2). Moreover, our forcing is by means of a wave maker driven with random noise within a frequency band, and not by parametrically vibrated the vessel at a single frequency as in [18,19].

Thus, the possible spurious effects listed above are not responsible of the observation of the upscale wave generation that we ascribed to an inverse cascade mechanism. An indirect evidence of this mechanism is the following. If the system is now driven with a forcing at a higher frequency bandwidth in the capillary regime (e.g. f p ∈ [30, 35] Hz), no power law is observed on the wave spectrum at low frequency as expected by wave turbulence theory [START_REF] Nazarenko | Wave Turbulence[END_REF]. Indeed, capillary wave turbulence is assumed to be generated by a process involving 3-wave interactions whereas inverse cascade is predicted to occur only for an even number of interacting waves (e.g. the 4-wave interaction process of gravity wave turbulence) [START_REF] Nazarenko | Wave Turbulence[END_REF].

Upscale transfer cut-off. -Let us now focus on the low frequency cut-off of the power law spectrum of the gravity regime. A peak centered at 3.7 Hz is observed in the low-frequency part of the spectrum (maximum amplitude of the spectrum in Fig. 2), that seems to end the power law. This frequency corresponds to a wavelength close to half of the vessel size, L/2. Indeed, by using the dispersion relation of gravity waves ω(k) = √ gk tanh kh, the resonant frequencies of the vessel corresponding to wavelengths 2L, L and L/2 are respectively 1, 2 and 3.6 Hz. The modes corresponding to wavelengths L and L/2 are displayed in Fig. 2 (see vertical dashed lines), and well correspond to peaks observed on the spectrum. The peak centered at 3.7 Hz is thus related to a resonant frequency mode of the vessel that fixes the lowest accessible frequency of the power-law gravity spectrum. The low frequency cut-off of the inverse cascade is thus due to the horizontal finite size of the experimental setup. To what extent this cut-off is related to a wave condensation phenomenon (large scales generated by the inverse cascade accumulate in a fundamental mode [11]) remains an open question. Finally, note that the spectrum amplitude at the resonant frequency f L/2 increases with the driving but does not broaden in frequency (see Fig. 2). This rules out a possible mechanism that would transfer energy directly from this fundamental mode to smaller scales instead of the suggested mechanism of upscale wave generation by an inverse cascade of wave action.

Spectrum scaling with the injected power. -We now turn to the scaling of the spectrum amplitude S η (f ) with the mean injected power I . Figure 3 shows the wave spectra S η (f ) rescaled by I for different I (varying by a factor 6) and for a random forcing at intermediate scales (f p ∈ [16, 22] Hz). For frequencies f < f p , all spectra are found to collapse. The best scaling for this regime is S η ∼ I 1±0.1 . Surprisingly, this scaling for the inverse regime of gravity wave turbulence is the same as the one obtained for both gravity and capillary direct cascades when using a large scale forcing [7,18]. We have checked that the immersion height of the wave maker H does not change this scaling of both direct cascades for the large scale forcing. Finite size effects of the container could be an explanation for this similar scaling for both inverse and direct gravity wave regimes: inverse transfers generate larger and larger scales up to almost the vessel size leading to a significant finite size effect, as when the forcing is at large scale since it directly generates scales of the order of the vessel size. The inverse cascade of gravity waves theoretically depends on the mean wave action flux q ≡ q(k)dk and its spectrum is predicted to scale as q1/3 (see above). Although the mean energy flux ε ≡ ε(k)dk is a measurable quantity (ε ∼ I -see [7]), and q(k) and ε(k) are related by ∂ε(k) ∂k = ω(k) ∂q(k) ∂k , we have currently no way to measure q. An estimate of the value of q would require an estimate of the number of wave modes in interaction. It is not possible with a temporal measurement at a single location but should deserves further studies with a spatio-temporal one [20].

Distribution of the injected power. -Figure 4 shows the probability density function (PDF) of the instantaneous injected power, I(t), rescaled by its rms value, σ I , in the presence of an inverse cascade (intermediate scale forcing) or in the presence of a direct gravity cascade (large scale forcing) both for the same forcing value of I . In both cases, the main properties of the PDF found in [21] are observed: i) the most probable value is 0, ii) strong fluctuations occur up to 10 times the rms value σ I , iii) the PDF displays roughly exponential tails for both positive and negative values of I, and iv) the model introduced in Ref. [21] with any adjustable parameter (when I /σ V σ F ≃ I /σ I is experimentally known) captures well both PDF shapes (see dashed lines in Fig. 4). Moreover, we find that both PDFs do not depend on the forcing amplitude. Indeed, the PDF(I/σ I ) for different forcing (within our range of I ) well collapse either for the direct cascade case or for the inverse one (not shown).

The main difference between both PDFs is that negative events of injected power are more likely when upscale transfer occurs than with only direct gravity cascade. It means that the energy coming back to the wave maker is more probable than for a direct cascade leading to a more symmetric PDF. This increase of energy transferred to the wave maker can be seen as a large scale effective dissipation.

Conclusion. -Experiments related to the observation of an inverse cascade in wave turbulence are scarce. Here, we have performed experiments in order to observe in laboratory the inverse cascade in gravity wave turbulence on the surface of a fluid. To wit, surface waves are forced at an intermediate scale corresponding to the gravity-capillary wavelength. Due to nonlinear wave interactions, waves at larger scales are observed and their amplitudes exhibit a frequency-power law spectrum at high enough forcing. Both observations are ascribed to the upscale wave action transfers of gravity wave turbulence. The exponent of the spectrum is close to the predicted value of the weak turbulence theory. The amplitude of the spectrum is found to scale linearly with the mean energy flux. Finally, the measurement of the probability distribution of injected power fluctuations shows that the energy given back to the wave maker is more likely in the presence of upscale (inverse) transfers than in the presence of direct cascade in gravity wave turbulence. * * *

Fig. 1 :

 1 Fig. 1: (color online). Power spectrum density of wave amplitude for different increasing amplitudes of forcing (from bottom to top). Sinusoidal forcing fp = 19 Hz. Wave maker immersion H = 10 mm. Curves are shifted vertically for clarity by a factor 1, 10, 100 and 1000 (from bottom to top). δω = 3.2 Hz (see text).

Fig. 2 :

 2 Fig. 2: (color online). Power spectrum density of wave amplitude for different forcing amplitudes: I = 2 mW (lower curve), I = 15 mW (medium curve) and I = 45 mW (upper curve). Slopes are: -3.3 (dashed line) and -3.6 (dot-dashed line). Vertical dashed-lines: theoretical resonant frequencies of the vessel fL = 2 Hz and f L/2 = 3.6 Hz. Maximum amplitude of the spectrum centered at 3.7 Hz is close to the vessel mode f L/2 . Random forcing: 16 ≤ fp ≤ 22 Hz (thick arrow). Wave maker immersion H = 10 mm. Vessel size: L = 20 cm.

Fig. 3 :

 3 Fig. 3: (color online). Power spectrum density of wave amplitude rescaled by the mean injected power Sη/ I . Different forcing amplitudes 10 ≤ I ≤ 60 mW and wave maker immersions H = 9, 10 and 11 mm. Slope of dashed line is -3.3. Random forcing: 16 ≤ fp ≤ 22 Hz.

Fig. 4 :

 4 Fig. 4: (color online). PDFs of the rescaled injected power I/σI . Red (light gray) solid line: with an inverse gravity cascade (intermediate scale forcing fp ∈ [16, 22] Hz). Black solid line: with a direct gravity cascade (large scale forcing fp ∈ [0.1, 6] Hz). I = 45 mW for both cases. Dashed lines are the predictions of Ref. [21] with I /σI = 0.15 [red (light gray)], and 0.5 (black). Dashed-dotted lines are I /σI = 0.15 [red (light gray)], and 0.5 (black). H = 10 mm.

  ). PDFs of the rescaled injected power I/σI . Red (light gray) solid line: with an inverse gravity cascade (intermediate scale forcing fp ∈ [16, 22] Hz). Black solid line: with a direct gravity cascade (large scale forcing fp ∈ [0.1, 6] Hz). I = 45 mW for both cases. Dashed lines are the predictions of Ref. [21] with I /σI = 0.15 [red (light gray)], and 0.5 (black). Dashed-dotted lines are I /σI = 0.15 [red (light gray)], and 0.5 (black). H = 10 mm.

We thank M. Berhanu for fruitful discussions. This work has been supported by ANR Turbonde BLAN07- 3-197846.


Carolina V S Borges

Pérola Milman

Arne Keller

Bell inequalities with continuous angular variables

Keywords: numbers: 03.65.Ud, 03.67.-a, 33.20.Sn

We consider bipartite quantum systems characterized by a continuous angular variable θ ∈ [-π, π[, representing, for instance, the position of a particle on a circle. We show how to reveal non-locality on this type of system using inequalities similar to CHSH ones, originally derived for bipartite spin 1/2 like systems. Such inequalities involve correlated measurement of continuous angular functions and are equivalent to the continuous superposition of CHSH inequalities acting on bidimensional subspaces of the infinite dimensional Hilbert space. As an example, we discuss in detail one application of our results, and we derive inequalities based on orientation correlation measurements. The introduced Bell-type inequalities open the perspective of new and simpler experiments to test non locality on a variety of quantum systems described by continuous variables.

Introduction:

The pioneering discussions of Einstein, Podolsky and Rosen (EPR) [1], which rose the possibility of eventual conflict between the classical an the quantum definitions of realism and locality, dealt with measurements of continuous variables (CV) of a quantum system, as position q and the kinetic momentum p. Almost 30 years elapsed before J. S. Bell promoted a regain of interest on the subject, deriving a quantitative criteria establishing the border between quantum and classical physics concerning locality and realism [2]. The discretized version of the EPR paradox and of the Bell inequalities using spin-1/2 like systems came with the work of Bohm [3] and Clauser, Horne, Shimony and Holt (CHSH) [4] and was motivated by the necessity to find simpler setups for experimentally testing the, up to then, Gedankenexperiments exposing the quantum-classical contradiction [5]. The almost 30 years following such pioneering experiments led to theoretical and experimental advances pointing out that using CV may provide advantages with respect to discrete systems. Higher intensity signals allowing to circumvent the detection loophole are among the announced advantages [6,7]. Most of these works are based on canonical unbounded variables, as (q, p). They can correspond, for instance, to the sum and the difference of two quadratures of the electromagnetic field. The first attempts to build CV non-locality tests consisted on finding ways to discretize the continuum to reuse concepts developed for the discrete case. It was shown in [8] that dichotomizing the phase space according to a state's parity and its displacement in phase space can lead to Bell type inequalities that can be violated by gaussian continuous variable entangled states. Other phase space dichotomizations are possible, as the one proposed in [9]. However, dichotomization does not correspond to a genuine CV measurement, since it provides an observable with a discrete, rather than genuinely continuous spec-trum.

Reid and co-workers [7] proposed variance based Belltype inequalities not demanding dichotomization, that are independent of the spectrum of the measured observables. Another possible approach are entropy based inequalities that may be advantageous to detect entanglement in non-gaussian states, since they go beyond the usual criteria that use second-order moments [10,11].

Up to now, most of the results on CV entanglement and non locality have been devoted to the case of canonical variables with an unbounded spectrum, as position and momentum. In the present work, we consider a different type of CV. We deal here with two quantum systems, A and B, characterized by angular variables θ i ∈ [-π, π[, i = A, B on a circle, instead of a position q in the full real line. θ can represent, for instance, the position of a particle on a circle, a rotator on a plane (for example, molecules confined to a plane [12]), or the polar angle locating the photon field in the plane transverse to its propagation, as illustrated in Fig. 1. The main result of the present paper is to build inequalities involving correlated measurement of continuous, real and periodic angular functions f i (θ i ) (i = A, B). Surprisingly, the derived inequalities share similarities to CHSH-like ones. As a matter of fact, their general form is given by a properly balanced continuous superposition of CHSHtype inequalities. Exploiting the possibility of revealing non locality using angular measurements is an approach that can be particularly useful and suitable for a number of physical systems in atomic and molecular physics, photonics and mesoscopic systems in condensed matter physics. √ 2 [13]. We now consider a different situation of measurement providing continuous and 2π-periodic outcomes. In other words, the underlying Hilbert space is the square integrable 2π-periodic functions space (L 2 ([-π, π[, dθ)). Such functions can be spanned by the basis {|m i ; m i ∈ Z}, which are angular momentum eigenstates: J z |m i = m|m i . An alternative continuous basis is {|θ i ; θ i ∈ [-π, π[}, and it can be obtained from the |m i basis by Fourier transform:

m i |θ i = 1 √ 2π e imiθi .
In this representation, a local observable for each party can be written as

F i = π -π dθ i f i (θ i )|θ i θ i |, with f i (θ i ) real, bounded and periodic.
Our guidelines to obtain continuous variables Bell inequalities is to build a CHSH [4] Bell-operator similar to the one given by ( 1), but based on the correlated measurements of F i for each particle. It is clear that, under the assumption that the spectra of F i are bounded, this property is preserved by every unitary transformation

U i (φ i ) such that F (φ i ) = U i (φ i )F i U † i (φ i ).
It is also straightforward to show that, for a LHV theory, we have:

| B(φ a , φ a , φ b , φ b ) | = | F A (φ a ) ⊗ F B (φ b ) + F A (φ a ) ⊗ F B (φ b ) +F A (φ a ) ⊗ F B (φ b ) -F A (φ a ) ⊗ F B (φ b ) | ≤ 2, (2)
if the maximum value of f i (θ i ) is normalized to 1. What are the conditions the functions f i (θ i ) and the transformed operators F i (φ i ) should satisfy to violate (2) and allow for non-locality and entanglement tests? In order to answer this question, we focus on f i (θ i ) = cos(θ i ) which is a function that can be measured on a variety of physical systems. The results derived in the following can be straightforwardly generalized to all 2π periodic functions such that f

(θ) = -f (θ -π) ∀θ ∈ [0, π].
Example of application: The observables C = F i corresponding to f i (θ i ) = cos(θ i ), can be expressed on the |m basis as:

C = m∈Z 1 2 (|m + 1 m| + |m m + 1|) . (3)
For a particle rotating on a circle, this operator corresponds to the projection on the x axis of the particle position; it can also be related to the spatial orientation of a two dimensional rotor, or to the phase of a superconducting circuit. Operator C has a spectrum of doubly degenerated eigenstates |θ and |-θ , both with eigenvalue cos θ. For this operator, we can define an equivalent to the rotation of a spin 1/2 system: it is the unitary operator e iJ 2 z φ , (that is, the free evolution operator during a time t = 2 φ, for a free particle with unit mass and angular momentum J z , constrained to move on an unit circle). From C, we can define C(φ i ) ≡ F i (φ i ) as:

C(φ) = e iJ 2 z φ Ce -iJ 2 z φ = 1 2 m∈Z e i(2m+1)φ |m m + 1| + e -i(2m+1)φ |m + 1 m|, (4)
and for a bipartite system (particles A and B) the Bell operator (2) reads: As a first approach to the study of the dependence of operators B(ξ a , ξ b ) with the phases ξ i , we discretize the possible outcomes of the correlation measurements. For this purpose we consider that the measured quantum states lie on a 2M +1 finite dimension space generated by the basis set {|m ; m ∈ Z, |m| ≤ M }. We thus define the projection C (M ) (ξ) of the C(ξ) operator on this reduced space as :

B(φ a , φ a , φ b , φ b ) = C A (φ a ) ⊗ C B (φ b )+C A (φ a )⊗C B (φ b )+C A (φ a )⊗C B (φ b )-C A (φ a )⊗ C B (
C (M ) (ξ) = 1 2 |m|≤M e i(2m+1)ξ |m m+1|+e -i(2m+1)ξ |m+1 m|
(5) and C (M ) = C (M ) (0). We also define discretized Bell operators B (M ) and B (M ) , analogously to the continuous case, but where the C(ξ i ) operators are replaced by C (M ) (ξ i ) operators, which act on the considered finite dimensional space only. Discretization allows to explicitly compute the 2M +1 eigenstates and corresponding eigenvalues of C (M ) , providing thus a more intuitive physical image of the considered operators. Also, it enables the numerical study of the spectrum of B (M ) (ξ a , ξ b ) and The study of the discretized operators is now used as a starting point to the derivation of the main results of this Letter, which are non-locality and entanglement tests in the continuous limit. We start by analyzing the point ξ a = ξ b = π/2, since it is the one that provides the highest contrast between b max (ξ a , ξ a ) and the violation threshold value (2).

We then move to the continuous limit (M → ∞) and search for the highest possible value of violation and the corresponding non local states for this choice of phases. Using the expression of C(ξ) (4) and the anti-symmetric property of cos θ, we compute θ |C( π 2 )|θ explicitly:

C( π 2 ) = π 0 dθ cos θ [i|θ -π θ| -i|θ θ -π|] . (6)
It is now convenient to define the following operators :

σ x (θ) = |θ θ| + |θ θ| σ y (θ) = i |θ θ| -|θ θ| (7) σ z (θ) = |θ θ| -|θ θ|,
for θ ∈ [0, π[ and with θ ≡ θ -π. These operators constitute an orthogonal continuous set of Pauli-like operators, and they fulfill relations analog to the usual Pauli matrices: σ j (θ)σ k (θ ) = iσ l (θ)δ(θ -θ ) where (j, k, l) denotes circular permutations of (x, y, z). They also fulfill orthogonality relations σ j (θ)σ j (θ ) = δ(θ -θ )(|θ θ| + θ θ ).

Eq.( 6) can thus be recast as:

C = π 0 dθ cos θ σ z (θ) (8)
C( π 2 ) = - π 0 dθ cos θσ y (θ). (9)
Thus, the Bell operator B m ≡ B( π 2 , π 2 ) can be written as the following direct sum:

B m = π 0 π 0 dθdθ cos θ cos θ X(θ, θ ) (10)
where

X(θ, θ ) = [σ z A (θ) ⊗ σ z B (θ ) -σ y A (θ) ⊗ σ z B (θ ) -σ z A (θ) ⊗ σ y B (θ ) -σ y A (θ) ⊗ σ y B (θ )] , (11)
and σ j A (θ) and σ j B (θ) (j = x, y, z) are operators defined by Eqs. (7) on the Hilbert space of parties A and B respectively. The X(θ, θ ) operators are orthogonal:

X(θ a , θ b )X(θ a , θ b ) = 4δ(θ a -θ a )δ(θ b -θ b )× (|θ a θ a | + θ a θ a ) ⊗ (|θ b θ b | + θ b θ b ) (12)
and completely analog to the usual 2-qubits (4dimensional) CHSH operators. We have thus the surprising result that the Bell operator B m is the weighted continuous direct sum of 2-qubit-like CHSH Bell operators X(θ, θ ), with the weights being given by cos θ cos θ . Thanks to the orthogonality property (12), finding the spectrum and eigenstates of B m is a simple task. Indeed, for each θ and θ it is enough to diagonalize the 4 × 4 matrix representing X(θ, θ ). We find that B m can thus be written in diagonal form as:

B m = 2 √ 2 n=±1 n π 0 π 0 dθdθ × cos θ cos θ |χ n (θ, θ ) χ n (θ, θ )| (13)
where : In order to explicitly construct physically sound states violating our Bell inequality, we should consider wave packets consisting of continuous superposition of eigenstates |χ +1 (θ, θ ) , with θ and θ localized around θ = θ = 0, point of the maximum eigenvalue of B m . An example of such wave packet is given by:

|χ ±1 (θ, θ ) = 1 N ± [|θ ⊗ |θ + |θ ⊗ |θ ∓ i( √ 2 ∓ 1) |θ ⊗ |θ + |θ ⊗ |θ ] (14
|Ψ = π 0 dθ π 0 dθ g a (θ)g b (θ )|χ +1 (θ, θ ) . ( 15
)
where g a (θ) and g b (θ) are normalized L

taking the same coefficients of Eq. ( 14):

|Ψ = 1 √ 2N + [(|g a ⊗ |g b + |g a ⊗ |g b ) (18) -i( √ 2 -1) (|g a ⊗ |g b + |g a ⊗ |g b )].
It is useful now to obtain a simple relation between the wave packet localization and the violation of the Bell inequality. For such, we take for g(θ) the ideal case of an angular slit with aperture δθ, given by :

g a (θ) = g b (θ) = 1 √ δθ for θ < δθ 0 otherwise (19)
Using Eq. ( 16), the value of B m Ψ can be written as function of the aperture δθ of the slit as follows:

B m Ψ = 2 √ 2 sin δθ δθ 2 . ( 20
)
This equation above shows that we can obtain values of B m Ψ > 2, violating the Bell inequality, with an aperture of δθ < 18 • which is not a too restrictive condition.

There is thus a relatively broad collection of simple twoparticles non local pure states involving coherent superposition of one-particle wave packets localized around θ = 0 and θ = π that violate the derived Bell inequality. We now study the example of even more realistic states, which are non-pure ones, establishing some conditions for them to violate the derived Belltype inequalities.

We consider the analog of the Werner states [16] : [17], where ρ A(B) = Tr B(A) [|Ψ Ψ|] = Conclusion: we derived general CHSH-like Bell-type inequalities for continuous variables using measurements of bounded observables with anti-symmetric and periodic spectra. Such observables are relevant for a number of experimental systems, ranging from the photonic orbital angular momentum to the movement of material particles on a circle and the phase of superconducting currents. We discussed in detail an example of observable, but the results obtained are valid for all observables with the same spectral properties and symmetries. Our results exploit angular variables from an original approach, and open the path to novel experiments with a wide range of quantum devices, throwing some light on the difficult task of non-locality violation with genuinely continuous variables.

ρ(η) = ηρ A ⊗ ρ B + (1 - η) |Ψ Ψ|

  Theoretical model: As a starting point, we recall the CHSH inequalities for a bipartite two-level system. It can be written in a simple form as | B(a, a , b, b ) | ≤ 2, with: B(a, a , b, b ) = σ a ⊗σ b +σ a ⊗σ b +σ a ⊗σ b -σ a ⊗σ b (1) where σ α is the Pauli matrix in one direction α of the tridimensional space. a and a refer to different directions of the spin projection of subsystem A while b and b are associated to subsystem B. The inequality | B(a, a , b, b ) | ≤ 2 is fulfilled in the framework of local hidden variable (LHV) theories. The fact that there exist quantum states violating these inequalities disproof quantum mechanics as a local or EPR realistic theory. For quantum mechanical systems, (1) is bounded by the Tsirelson bound of 2

  φ b ), where C A(B) (φ) are operators defined as in Eq. (4) acting on the Hilbert space of particle A(B). The spectrum of C(φ) does not depend on φ. Diagonalizing C(φ) shows that its spectrum is bounded, with | C | ≤ 1. Thus, for a LHV theory, | B(φ a , φ a , φ b , φ b ) | < 2 holds. However, for some set of phases φ i s and some quantum states, this inequality is violated. In order to show that, we calculate the spectra of B(φ a , φ a , φ b , φ b ) operators. Actually, such spectra depend only on the relative phases φ a -φ a and φ b -φ b[14,15]. We can defineB(ξ a , ξ b ) = B(φ a -φ a , φ b -φ b ) ≡ B(0, φ a -φ a , 0, φ b -φ b ) = e iJ 2 Az φa ⊗ e iJ 2 Bz φ b B(φ a , φ a , φ b , φ b )e -iJ 2Az φa ⊗e -iJ 2 Bz φ b . B(ξ a , ξ b ) and B(φ a , φ a , φ b , φ b ) are related by an unitary transformation. Therefore, the variation of only the two phases (ξ a , ξ b ) is enough to explore the spectrum of all the B(φ a , φ a , φ b , φ b ).

FIG. 1 .

 1 FIG. 1. Examples of physical systems for which our results can be applied: (a) The transverse profile of a propagating light beam (b) Particles moving on a circle (as charged particles in an electromagnetic trap) (c) rigid rotators confined to two dimensional motion.

2 ) 1 / 2 2 , 2 √ 2 ]

 212222 ) are the eigenvectors of X(θ, θ ) and of B m with non-zero eigenvalues and where N ± = 2(2 ∓ √ is a normalization factor such that :χ n (θ a , θ b )|χ n (θ a , θ b ) = δ nn δ(θ a -θ a )δ(θ b -θ b ).Therefore, the spectrum of B m is continuous and equal to [-2√ . | B m | is thus bounded by 2 √ 2 as in the 2-qubit CHSH inequality. However, the values ±2 √ 2 are only reached by eigenstates of B m which are not physical states since they are composed of perfectly oriented states, namely |0 and |π . In the angular representation, these states are expressed by delta functions, since θ|θ = δ(θ -θ ).

2 ( 16 )

 216 2 (]0, π[, dθ) functions with support containing θ = 0. The expectation value of B m for this state is B m θ cos θ |g a (θ)| 2 |g b (θ )| The wave packet (15) can be produced making a linear combination of the one-particle wave packets |g = π 0 dθg(θ)|θ and |g = π 0 dθg(θ)|θ ,

1 2 g 1 √ 2 δθ sin δθ 2 ,

 12122 a(b) g a(b) + g a(b) g a(b) , with |Ψ given by (18). Asχ n (θ a , θ b )|ρ A ⊗ ρ B |χ n (θ a , θ b ) does not depend on n = ±1 for all θ a , θ b ∈ [0, π], then Tr [B m ρ A ⊗ ρ B ] = 0.It turns out that the expectation of B m for the state ρ(η) is simply given by (1 -η) B m Ψ . In the ideal case where g a and g b are given by (19), the maximal allowed value of the mixing coefficient η for the Bell equation to be violated is a simple function of the slit aperture :Tr [B m ρ(η)] > 2 ⇒ η < 1 -providing condi-tions for this type of mixed states to violate non-locality.

Authors acknowledge funding by CAPES-COFECUB No. 640/09.


Caroline De Dieuleveult

Vincent Lagneau

Juan Hidalgo

Development of an isotopic fractionation and filiation module in the reactive transport code HYTEC

Development of an isotopic fractionation and filiation module in the reactive transport code HYTEC C. de Dieuleveult 1 , V. Lagneau 1 , J. J. Hidalgo 2,3 1 Centre de Geosciences, MINES ParisTech 2 Massachusetts Institute of Technology -CEC 3 Institute of Environmental Assessment and Water Research -CSIC The increasing place of numerical modelling in the geosciences, as in numerous fields is well known. Owing to rapid progress in computer sciences and hardware, more and more complex systems, i.e. also more and more realistic ones, can be simulated. HYTEC is a versatile coupled reactive transport code, currently used for several applications, such as groundwater pollution studies, safety assessment of nuclear waste disposals, CO 2 storage, geochemical studies or interpretation of laboratory column experiments. This work presents a newly developed functionality in HYTEC, which allows taking into account isotopic speciation and filiation. Variations in isotopic compositions give useful information for the geosciences. They can help track the evolution of aqueous and mineral species, mixing of different origin fluids, and are helpful for dating. Then, environmental isotopes experiences and modelling complete geochemical and physical hydrology studies.

For a specific element, all isotopes have similar chemical properties, due to an identical filling of their electronic orbitals. However, the slight difference in mass causes differences in reaction rates and equilibrium constants. This can be treated directly with as many reactions as isotopes, or more usually by first gathering all the isotopes within the element reaction then correcting with a fractionation factor. Moreover, isotopes can be radioactively unstable, leading to a chain of parent and daughter decay isotopes.

We have included in the reactive transport code HYTEC a description of isotopes, derived from chemical elements. Then, the radioactive filiation and isotopic fractionation were implemented. The overall resolution algorithm is based on an operator splitting method: the main idea is to take advantage of an analytical resolution of the filiation problem, which is the toughest to handle due to very large differences in half-lives, from milliseconds to millions of years. Between time t and t+dt, the algorithm can be decomposed in successive steps. First, HYTEC solves the filiation problem: the radioactive decay and filiation is computed analytically for all the isotopes. Second, HYTEC updates the total concentration for each element. Third, the chemical module updates the speciation. Fourth, a transport-chemistry iterative loop is performed: transport is calculated on the total aqueous concentration for each isotope, then isotope concentrations are collapsed for the element-based speciation resolution. Each time the speciation module is called, the fractionation resolution is performed to dispatch elementbased species concentrations into isotopic species concentrations.

The isotopic filiation model was tested on an simple problem with an analytical solution: the transport of Cesium 137 through a porous media containing a sorbing surface mineral. Three processes occur at the same time: transport of the isotopes, sorption of the Cesium on the surface sites and radioactive decay of 137 Ce into two daughters, 137 Ba and 137m Ba. The simulation results compared correctly with an analytical solution. A second reference is used to test the isotopic fractionation: carbon isotope partitioning as a function of pH and compared with results from the literature.

Work is continuing to improve the model. The resolution is based on the assumption of an overly predominant isotope, used as (user-defined) reference. The dominant isotope could change during the evolution of a system, or there could be two similar concentration isotopes. The first issue can be solved with a locally, automatically, adapted reference. The second requires an extension of the fractionation module. Finally, H and O are still a challenge due to the interference with the solvant.