Max Schemmer

Aisling Johnson

Isabelle Bouchoule

Monitoring squeezed collective modes of a 1D Bose gas after an interaction quench using density ripples analysis

Keywords: numbers: 03, 75, Hh, 67, 10, Ba II, THE INTERACTION QUENCH WITHIN

We investigate the out-of-equilibrium dynamics following a sudden quench of the interaction strength, in a one-dimensional quasi-condensate trapped at the surface of an atom chip. Within a linearized approximation, the system is described by independent collective modes and the quench squeezes the phase space distribution of each mode, leading to a subsequent breathing of each quadrature. We show that the collective modes are resolved by the power spectrum of density ripples which appear after a short time of flight. This allows us to experimentally probe the expected breathing phenomenon. Our results are in good agreement with theoretical predictions which take the longitudinal harmonic confinement into account.

I. INTRODUCTION

The out-of-equilibrium dynamics of isolated quantum many-body systems is a field attracting a lot of interest [START_REF] Polkovnikov | Colloquium: Nonequilibrium dynamics of closed interacting quantum systems[END_REF], triggered in part by progress in cold atom experiments. A particular focus has been devoted to the case of sudden quenches where the system is brought out-ofequilibrium by a sudden change of a Hamiltonian parameter, and in particular the case of an interaction quench, both theoretically [2] and experimentally [START_REF] Trotzky | Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas[END_REF][START_REF] Cheneau | Light-cone-like spreading of correlations in a quantum many-body system[END_REF][START_REF] Hung | From Cosmology to Cold Atoms: Observation of Sakharov Oscillations in a Quenched Atomic Superfluid[END_REF][START_REF] Langen | Double light-cone dynamics establish thermal states in integrable 1d Bose gases[END_REF][START_REF] Jaskula | Acoustic Analog to the Dynamical Casimir Effect in a Bose-Einstein Condensate[END_REF]. Whether and how the system relaxes towards an equilibrium state is the subject of intense theoretical work. The role of integrability, not completely elucidated, is the focus of many studies. Within this context, the case of a 1D Bose gas with contact repulsive interactions, described by the integrable Lieb-Liniger model, is a prime theoretical candidate to uncover the underlying physics, studied in e.g [START_REF] De Nardis | Solution for an interaction quench in the Lieb-Liniger Bose gas[END_REF][START_REF] Calabrese | Interaction quench in a LiebLiniger model and the KPZ equation with flat initial conditions[END_REF][START_REF] Cazalilla | Quantum quenches in the Luttinger model and its close relatives[END_REF][START_REF] Wisocki | Quantum fluctuation effects on the quench dynamics of thermal quasicondensates[END_REF].

This paper constitutes the experimental study of the out-of-equilibrium dynamics following a sudden quench of the interaction strength in a 1D Bose gas with repulsive interactions. Within a linearized approximation, the evolution following a splitting of a 1D Bose gas in two copies, studied in [START_REF] Langen | Double light-cone dynamics establish thermal states in integrable 1d Bose gases[END_REF], can be interpreted as an interaction quench in an effective 1D Bose gas. Investigating the first-order correlation function, the authors observed an apparent thermalization, taking the form of a light cone effect. This observation may however conceal underlying non-equilibrium dynamics, as revealed recently by the observation of recurrences in a similar experiment [START_REF] Rauer | Recurrences in an isolated quantum many-body system[END_REF]. Finding appropriate observables revealing these dynamics is thus a key point for investigating out-of-equilibrium phenomena. In this paper, by investigating the density ripples appearing after short time of flight, the behavior * current address: Vienna Center for Quantum Science and Technology, TU Wien-Atominstitut, Stadionallee 2, 1020 Vienna, Austria. † isabelle.bouchoule@institutoptique.fr

t = 0 - t = 0 + 0 1 2 ω q t/π (c)
FIG. 1. Squeezing of each collective mode after an interaction strength quench from gi to g f . The Gaussian phase space distributions before the quench (t = 0 -), just after the quench (t = 0 + ) and after an evolution time π/ (2ωq) (dashed ellipse) are represented in (b), where lines correspond to a given probability density (here we chose κ = 3). The subsequent breathing is seen in (c), where the time evolution of the phase distribution is shown in color plot.

of collective modes is probed, rather than a global quantity such as the first-order correlation function, allowing for a better understanding of the physics at play after an interaction quench. The dynamics is revealed by the oscillatory behavior of each component of the density ripples power spectrum, observed for times that go beyond the apparent thermalization time seen on the first order correlation function. We show that these oscillatory dynamics are the signature of squeezed collective modes: for each collective mode, the quench produces a squeezed phase space distribution, leading to a subsequent oscillation of the width of its quadratures -a breathing behavior. As well as improving the understanding of the effect of an interaction quench, this work constitutes an observation of squeezed collective modes, a result interesting on its own. and density n 0 , with particles of mass m interacting with a two-body repulsive contact interaction gδ(z), where z is the distance between the two particles. At t = 0, g is suddenly changed from g i to g f = (1+κ)g i , where κ is the quench strength. While the complete treatment of an interaction quench is tremendously difficult the problem is greatly simplified if one can rely on a linearized approach, as presented below. Within the quasi-condensate regime, density fluctuations are strongly reduced (|δn(z)| n 0 ) and phase fluctuations occur on large length scales, such that the Hamiltonian of the system can be diagonalized using the phase-density representation of the field operator Ψ(z) = n 0 + δn(z) exp(iθ(z)) and the Bogoliubov procedure [START_REF] Mora | Extension of Bogoliubov theory to quasicondensates[END_REF]. The obtained linearized modes correspond to Fourier modes. For each wave-vector q, the dynamics is governed by the harmonic oscillator Hamiltonian [START_REF] Schemmer | Monte Carlo wave-function description of losses in a one-dimensional Bose gas and cooling to the ground state by quantum feedback[END_REF]

H q = A q n 2 q + B q θ 2 q = ω q ñ2 q /2 + θ2 q /2 (1)
where the canonically conjugated hermitian operators n q and θ q are the Fourier components [START_REF]For each positive q value, one has 2 Fourier components: nq[END_REF] of δn and θ and where the reduced variables are defined by ñq = n q (A q /B q ) 1/4 and θq = θ q (B q /A q ) 1/4 . For wavevectors q much smaller than the inverse healing length ξ -1 = √ mgn 0 / , the excitations are of hydrodynamic nature [START_REF]For quasi-1D gases the hydrodynamic condition is replaced by ωq ω ⊥[END_REF]. Their frequency is ω q = cq, where the speed of sound is c = n 0 ∂ n µ/m, and the Hamiltonian's coefficients are B q = 2 q 2 n 0 /(2m) and A q = mc 2 /(2n 0 ).

Here µ(n) is the equation of state of the gas relating the chemical potential µ to the linear density, which reduces to µ = gn for pure 1D quasi-condensate. For a given q, the dynamics of the quenched harmonic oscillator is represented in Fig. [START_REF] Polkovnikov | Colloquium: Nonequilibrium dynamics of closed interacting quantum systems[END_REF]. Before the quench the phase space distribution is the one of a thermal state: an isotropic Gaussian in the ( θq , ñq )-plane. The quench affects A q while θ q and n q do not have time to change. The variances thus become θ2 q t=0 + = θ2 q t=0 -/(1 + κ) 1/2 and δ n q [START_REF]The phase space area is preserved, one quadrature being squeezed, while[END_REF]. The subsequent evolution is a rotation in phase space at a frequency ω q leading to a breathing of each quadrature. In particular

2 t=0 + = δ n q 2 t=0 -(1 + κ) 1/2
θ 2 q = θ 2 q i (1 + κ sin 2 (cqt)), (2)
where the initial value θ 2 q i is the thermal prediction

θ 2 q = mk B T /( 2 n 0 q 2 ) [18].
Probing the non equilibrium dynamics following a quench is not straightforward, especially concerning the choice of the observable. Since density fluctuations are very small within the quasi-condensate regime, it is more advantageous to probe the phase fluctuations [19]. One way is to investigate the one-body correlation function g 1 (z) = Ψ † (z) Ψ(0) , which, for z ξ and in the quasi-condensate regime, writes g 1 (z) n 0 e -(θ(z)-θ(0)) 2 /2 [START_REF] Mora | Extension of Bogoliubov theory to quasicondensates[END_REF]. However since phase fluctuations are large in a quasi-condensate, the exponential cannot be linearized and g 1 (z) mixes contributions from all Bogoliubov modes [START_REF]Isolating the contribution of individual modes to the function g1(z) requires looking at the Fourier transform of ln(g1(z)), which requires large detection dynamics[END_REF], preventing the observation of the squeezed dynamics. In fact, the linearized model above predicts the light-cone effect on the g 1 function: g 1 (z) changes from its initial exponential decay exp(-|z|/l i c ), where l i c = 2 2 n 0 /(mk B T ), to an exponential decay with a new correlation length l f c = 2l i c /(κ + 2) for z < 2ct. The breathing of each squeezed Bogoliubov mode is not transparent here. Moreover, for times larger than a few t g1 th = l f c /c, the g 1 function essentially reaches the form expected for a thermal state at a temperature T f = T (κ + 2)/2, and the ongoing dynamics is hidden. In this paper we use the density ripples analysis to reveal the non equilibrium dynamics of the gas by probing the breathing of each mode.

III. RESOLVING BOGOLIUBOV MODES WITH DENSITY RIPPLES

Density ripples appear after switching the interactions off and waiting for a free evolution time t f (timeof-flight), during which phase fluctuations transform into density fluctuations [21][START_REF] Dettmer | Observation of Phase Fluctuations in Elongated Bose-Einstein Condensates[END_REF][START_REF] Manz | Two-point density correlations of quasicondensates in free expansion[END_REF][START_REF] Rauer | Cooling of a One-Dimensional Bose Gas[END_REF]. Consider the power spectrum of density ripples |ρ(q)| 2 , where

ρ(q) = (1/ √ L) dz( n(z, t f ) -n 0 )e iqz .
Propagating the field operator during the time of flight and assuming translational invariance we obtain [START_REF]For consistency we rederive this expression[END_REF]

|ρ n0 (q)| 2 = dxe -iqx (f (q, x) -n 2 0 ), (3)
where

f (q, x) n 2 0 e i[θ(0)-θ(-qt f /m)+θ(x-qt f /m)-θ(x)] , (4)
averages in Eq. ( 4) are taken before the time of flight.

The function f involves only pairs of points separated by qt f /m. For small wave vectors q t f /m l c , the phase difference between those points is small and one can expand the exponential. To lowest order, assuming uncorrelated distributions for each mode q and vanishing mean values, we then find

|ρ n0 (q)| 2 = 4n 2 0 θ 2 q sin 2 q 2 t f /(2m) , (5)
showing that, for low lying q, the density ripples spectrum directly resolves the phase quadrature θ 2 q of individual Bogoliubov modes [START_REF]2 ,s )/2 where θq,c and θq,s are the cosine and[END_REF]. The proportionality between |ρ n0 (q)| 2 and θ 2 q implies that |ρ n0 (q)| 2 oscillates according to Eq. (2) when varying the time t after the quench. Density ripples are thus an ideal tool to investigate the quench dynamics. Note that, in the following we are interested, for each wave vector q, in the evolution of |ρ n0 (q)| 2 with the evolution time t, such that the proportionality factor 4n 2 0 sin 2 q 2 t f /(2m) is irrelevant for our data analyis.

In typical experiments, atoms are confined by a smooth potential V (z). For weak enough confinement and for wavelengths much smaller than the system's size, one can however use the above results for homogeneous systems within a local density approximation (LDA) [START_REF]Validity of LDA is established in Appendix E[END_REF]. Then ρ(q) = dzδn(z, t f )e iqz fulfills ρ(q) dz |ρ n0(z) (q)| 2 where n 0 (z) is the density profile, which can itself be evaluated within the LDA using the gas equation of state and the local chemical potential µ(z) = µ 0 -V (z). Injecting Eq. (2) and Eq. ( 5) into the LDA integral, we find

|ρ(q)| 2 / |ρ(q)| 2 i = 1 + κF(cqt), (6)
where c is the speed of sound after the quench evaluated at the trap center and F only depends on the shape of V (z). For a box-like potential, one recovers previous results and F(τ ) = sin 2 (τ ). The expression of F is given in Appendix D in the case of a harmonic potential: The oscillatory behavior is preserved, although the spread in frequencies originating from the inhomogeneity in n 0 introduces damping, which is a pure dephasing effect.

IV. EXPERIMENTAL REALIZATION

The experiment uses an atom-chip set up [28] where 87 Rb atoms are magnetically confined using currentcarrying micro-wires. The transverse confinement, acting in a vertical plane, is provided by three parallel wires carrying AC-current modulated at 400 kHz, which renders the magnetic potential insensitive to wire imperfections and, allows for independent control of the transverse and longitudinal confinements. We perform radio frequency (RF) forced evaporative cooling until we reach the desired temperature. We then increase the RF frequency by 60 kHz, providing a shield for energetic three-body collision residues and wait during 150 ms relaxation time. The clouds contain a few thousand atoms, in a trap with a transverse frequency ω ⊥ /2π = 1.5 or 3.1 kHz, depending on the data set, and a longitudinal frequency ω /2π = 8.5 Hz. The samples are quasi-1D, the temperature and chemical potential satisfying µ, k B T < ω ⊥ . The temperature is low enough so that the gas typically lies well within the quasi-condensate regime [START_REF] Kheruntsyan | Pair Correlations in a Finite-Temperature 1d Bose Gas[END_REF]. The equation of state is well described by µ = ω ⊥ ( √ 1 + 4na -1), where a = 5.3 nm is the 3D scattering length [START_REF] Fuchs | Hydrodynamic modes of a one-dimensional trapped Bose gas[END_REF]. While, for na 1, one recovers the pure 1D expression µ = gn, where g = 2 ω ⊥ a, this equation of state takes the broadening of the transverse size at larger na into account. The longitudinal density profile, well described by the LDA, extends over twice the Thomas-Fermi radius R T F = 2µ 0 /m/ω . The speed of sound derived from the equation of state is c = c 1D /(1 + 4na) 1/4 where c 1D = 2 ω ⊥ na/m is the pure 1D expression. For data presented in this paper, c/c 1D is close to 0.7. Since the effective interaction strength is proportional to c 2 , it is proportional to ω ⊥ .

The interaction strength quench is performed by ramping the transverse trapping frequency ω ⊥ from its initial value ω ⊥,i to its final value ω ⊥,f = (1 + κ)ω ⊥,i within a time t r , typically of the order of 1 ms. This time is long enough for the transverse motion of the atoms to follow adiabatically but short enough so that the quench can be considered as almost instantaneous with respect to the probed longitudinal excitations (see Appendix H 1). We simultaneously multiply ω by √ 1 + κ, to avoid modification of the mean profile and of the Bogoliubov wavefunctions(see Appendix E).

In order to probe density ripples, we release the atoms from the trap and let them fall under gravity for a time t f = 8 ms before taking an absorption image. The transverse expansion, occurring on a time scale of 1/ω ⊥ , ensures the effective instantaneous switching off of the interactions with respect to the probed longitudinal excitations. The density ripples produced by the phase fluctuations present before the free fall are visible in each individual image, as seen in Fig. (2)(a). From the image, we record the longitudinal density profile ρ(z, t f ) and its discrete Fourier transform [31] ρ(q). We acquire about 40 images taken in the same conditions with atom number fluctuations smaller than 10%. From this data set, we then extract the power spectrum |ρ(q)| 2 . We note |ρ(q)| 2 i the power spectrum obtained before the quench and a typical spectrum is shown in Power spectra after a quench of strength κ = 2, at times t = 2.1 ms (crosses, green), t = 2.6 ms (circles, blue) and t = 4.6 ms (squares, black), the solid (red) curve being the initial power spectrum.

predicted power spectrum |ρ(q)| 2 th is computed using the LDA and analytical solution of Eq. ( 3) for thermal equilibrium (see Appendices B, C). This expression is peaked around kR TF πm/( t f )R TF 50. For comparison with experimental data, we take the imaging resolution into account by multiplying |ρ(q)| 2 th with e -q 2 σ 2 where σ is the rms width of the impulse imaging response function, assumed to be Gaussian (Appendix F discusses the effect of this finite optical resolution). The experimental data ultimately compared well with the theoretical predictions, as shown in Fig. (2)(b), where T and σ are obtained by fitting the data [32] [33]. Finally we obtain k B T /µ 0 = 0.4, close to the lowest value obtained in similar setups [START_REF] Rauer | Cooling of a One-Dimensional Bose Gas[END_REF][START_REF] Jacqmin | Sub-Poissonian Fluctuations in a 1d Bose Gas: From the Quantum Quasicondensate to the Strongly Interacting Regime[END_REF].

We investigate the dynamics following the quench of the interaction strength by acquiring power spectra of density ripples at different evolution times t after the quench. We typically acquire power spectra every 0.5 ms, over a total time of 5 ms. A few raw spectra are shown in Fig. (2)(b), for a quench strength κ = 2.0. At first sight the power spectra seem erratic. In order to reveal the expected oscillatory behavior of each Fourier component we introduce, for each wavevector q of the discrete Fourier transform, and each measurement time t, the reduced time τ = cqt, where c is evaluated for the central density, and compute J(q, τ ) = |ρ(q)| 2 (t)/ |ρ(q)| 2 i . We restrict the range of q values to 10 < qR T F < 40, to ensure both the condition q t f /m l c and the validity of the LDA. On the resulting set of spare data, shown in the inset of Fig. [START_REF] Trotzky | Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas[END_REF], an oscillatory behavior appears, despite noise on the data. To combine all the data in a single graph, we perform a "smooth" binning in τ , i.e. we compute, for any given reduced time τ , the weighted averaged of the data with a Gaussian weight function in τ of width ∆ = 0.31 : namely we compute J(τ ) =

α J(q α , τ α )e -(τα-τ ) 2 /(2∆ 2 ) / α e -(τα-τ ) 2 /(2∆ 2 )
, where the sum is done on the data set. The function J(τ ), shown in Fig. (3) shows a clear oscillatory behavior.

We repeat the experiment for different quench strengths κ = (ω ⊥,f /ω ⊥,i -1) = {0.3, 3, 5}, and initial trapping oscillation frequencies ω ⊥ = {3, 1.5} kHz. The oscillatory behavior is present in all cases as shown in Fig. [START_REF] Trotzky | Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas[END_REF]. We compared the observed oscillations with the theoretical predictions from the linearized model, Eq. ( 6). The temporal behavior of the data is in good agreement with the predicted one: both the frequency and the observed damping are in agreement with the predictions. The amplitude of the experimental oscillations on the other hand are significantly smaller than the predictions, and in Fig. (3) we plot the theoretical predictions for quench strengths twice as small as the experimental ones. Moreover, for a given quench strength, the observed amplitude depend on the initial transverse frequency, in contradiction with the theoretical model. Several effects leading to a decrease of the oscillation amplitude are discussed in Appendix H. However, they only partially account for the observed amplitude reduction. Inset shows the data corresponding to each measurement time and discrete q values, for a data set corresponding to κ = 2 and ω ⊥,i = 2π × 1.5 kHz, together with the resulting continuous averaged quantity J (see text). Orange crosses correspond to t < t g 1 th and blue circles to t > t g 1 th . The main graph shows the evolution of the experimental smoothed quantity J for different data sets. The error bars show the typical statistical uncertainty on J. The initial transverse oscillation frequency is 1.5 kHz, except for the thick dark grey (blue) curve for which it is 3 kHz. Quench strengths are κ = 4 (light gray (orange)), κ = 2 (dark gray (blue)) data and κ = -0.7 (black). Dashed lines are theoretical predictions for quench strengths κ = 2 (lightgray (orange)), 1 (light gray) and -0.35 (black).

V. DISCUSSION

In conclusion, analyzing density ripples, we revealed the physics at play after a sudden quench of the interaction strength in a quasi-1D Bose gas, namely the breathing associated to the squeezing of each collective mode. The observed out-of-equilibrium dynamics continues for times larger than t g1 th , for which the g 1 function essentially reached its asymptotic thermal behavior [35] This can be seen in the inset of Fig. [START_REF] Trotzky | Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas[END_REF] where data corresponding to t > t g1 th , shown in blue circles, still present an oscillatory behavior. This clearly underlines the power of the density ripple analysis to unveil out-of-equilibrium physics. The observed damping is compatible with the sole dephasing effect due to the longitudinal harmonic confinement. At later times, the discreteness of the spectrum and its almost constant level spacing is expected to produce a revival phenomenon. Its observation might however be hindered by a damping of each collective mode due to non-linear couplings. Such a damping occurs, despite the integrability of the 1D Bose gas with contact repulsive interactions, because the Bogoliubov collective modes do not correspond to the conserved quantities. A long-lived non-thermal nature of the state produced by the interaction strength might be revealed either by observing excitations in both the phononic regime and the particle regime of the Bogoliubov spectrum [START_REF] Johnson | Long-lived nonthermal states realized by atom losses in one-dimensional quasicondensates[END_REF], or, ideally, in finding a way to access the distribution of the Bethe-Ansatz rapidities.

The power spectrum of density ripples has been first investigated in the limit of small density ripples and for a gas initially in the 3D Thomas-Fermi regime (i.e. µ ω ⊥ ) [START_REF] Dettmer | Observation of Phase Fluctuations in Elongated Bose-Einstein Condensates[END_REF][START_REF] Hellweg | Phase fluctuations in BoseEinstein condensates[END_REF]. It was then computed assuming instantaneous switching off of the interactions in [21]. Here, for consistency, we rederive Eq. ( 4) and ( 5) of the main text. Since we will later consider trapped gases, let us first assume a general scenario where we do not restrict ourselves to the homogeneous case. We let the gas evolve freely for a time t f after interactions have been switched off. The power spectrum of the density fluctuations after t f writes

|ρ(q)| 2 = dz 1 dz 2 e iq(z1-z2) δn(z 1 , t f )δn(z 2 , t f ) .
(A1) Writing δn(z) = n(z)n(z) and expanding the above equation, the term | dze iqz n(z, t f ) | 2 appears. Here we consider times of flight short enough so that the shape of the cloud barely changes during time of flight, so that n(z, t f ) n(z, 0) . We moreover consider wavevectors q much larger than the inverse of the cloud length, such that | dze iqz n(z, 0) | 2 is a negligible quantity. We then have

|ρ(q)| 2 dz 1 dz 2 e iq(z1-z2) n(z 1 , t f )n(z 2 , t f ) . (A2) To compute n(z, t f ) = Ψ + (z, t f )Ψ(z, t f )
we evolve the atomic field with the free-particle propagator, which leads to

ψ(z, t f ) = 1 2πt f dαψ(α, 0)e i (z-α) 2 2t f , ( A3
)
where for simplicity we use a unit system in which m = = 1. We then have

n(z 1 , t f )n(z 2 , t f ) = 1 (2πt f ) 2 dαdβdγdδ ψ + α ψ β ψ + γ ψ δ e -i (z 1 -α) 2 2t f e i (z 1 -β) 2 2t f e -i (z 2 -γ) 2 2t f e i (z 2 -δ) 2 2t f , ( A4
)
where we use the simplified notation ψ ν = ψ(ν, 0). Expanding the exponentials, the above expression writes

n(z 1 , t f )n(z 2 , t f ) = 1 (2πt f ) 2 dαdβdγdδ ψ + α ψ β ψ + γ ψ δ e i (α-β)z 1 t f e i β 2 -α 2 2t f e i (γ-δ)z 2 t f e i δ 2 -γ 2 2t f . (A5)
Injecting into Eq. (A2), and using dze ikz = 2πδ(k) and δ(x/α) = αδ(x), we get

|ρ(q)| 2 = dαdδ ψ + α ψ α+qt f ψ + δ+qt f ψ δ e -i α 2 2t f e i (α+qt) 2 2t f e -i (δ+qt) 2 2t f e i δ 2 2t f . (A6) Defining δ = α + X, we obtain |ρ(q)| 2 = dαdXe iqX ψ + α ψ α+qt f ψ + α+X+qt f ψ α+X . ( A7
)
For gases lying deep in the quasi-condensate regime, one can neglect density fluctuations when estimating the expectation value in the above equation, such that

|ρ(q)| 2 dαdXe iqX n(α)n(α + qt f )n(α + X + qt f )n(α + X) e i(θ(α)-θ(α+qt f )+θ(α+X+qt f )-θ(α+X)) . (A8)
The following section applies this result to homogeneous systems. This equation is however not restricted to homogeneous systems and we will use it to treat the effect of the trap beyond the local density approximation.

Appendix B: Power spectrum of the density ripples for a homogeneous gas

For a homogeneous gas, the relevant quantity is an intensive variable which relates to the expression |ρ(q)| 2 of the previous section by

|ρ(q)| 2 = 1 L |ρ(q)| 2 (B1)
where L is the length of the box. Injecting Eq. (A8) into Eq. (B1), we recover Eq. ( 3) and (4) of the main text, up to an irrelevant term in δ(q) [39]. In fact, Wick's theorem is applicable since θ is a Gaussian variable [40], which leads to

|ρ(q)| 2 n 2 0 = dXe iqX-1 2 (θ(0)-θ(qt f )+θ(X+qt f )-θ(X)) 2 .
(B2) To compute the power spectrum of density ripples for a thermal equilibrium state, we follow the calculation made in [21] and expand the exponential term in Eq. (B2) as a function of the first order correlation function g (1) (z) = n 0 e -1 2 (θ(0)-θ(z)) 2 , which fulfils g (1) (z) = n 0 e -|z|/lc where l c = 2 2 n 0 /(k B T ) [21]. Calculation of the integral in Eq. (B2) then leads to

|ρ(q)| 2 n 2 0 = 4ql c q(4 + l 2 c q 2 ) - 4e -2 qt f mlc ql c cos( q 2 t f m ) + 2 sin( q 2 t f m ) q(4 + l 2 c q 2 )
. (B3) FIG. 4. Density ripples power spectrum for a homogeneous gas. The exact formula Eq. (B3) (dashed curve) is compared to the small q approximation given Eq. ( 5) of the main text, where |ρ(q)| 2 is proportional to θ 2 q (solid curve). The only relevant parameter is t f /(ml 2 c ). Results are shown for t f /(ml 2 c ) = 0.05, a value corresponding to the data depicted in Fig. (2,b) of the main text, the correlation length lc = 2 2 n0/(mkBT ) being computed for the central density. The effect of the imaging resolution is to multiply this theoretical power spectrum with e -σ 2 q 2

, where σ is the rms width of the imaging pulse response function, assumed to be Gaussian. For our data, σ m/( t f ) = 0.85 and only the first maximum of |ρ(q)| 2 remains visible.

Note that we corrected the formula given in [21]. The power spectrum computed with this equation is compared in Fig. 4 to the approximated formula valid for small q, namely Eq. ( 5) of the main text.

Appendix C: Density ripple power spectrum for a harmonically confined gas under the LDA Let us investigate the density ripples power spectrum in the case of a gas trapped in a longitudinal potential smooth enough so that the cloud size L is much larger than the typical phase correlation length l c and much larger than qt f /m: L l c , qt f /m. As in section A, we moreover consider the power spectrum for wavevectors q 1/L. Let us start with the general expression Eq. (A1) that we write |ρ(q)| 2 = dz du δρ(z, t f )δρ(z + u, t f ) e iqu . (C1) Consider δρ(z, t f )δρ(z + u, t f ) for a given z. This expression vanishes over a length much smaller than L, so values of u significantly contributing to the integral are much smaller than L. Moreover the region of the initial cloud contributing most to δρ(z, t f )δρ(z+u, t f ) is much smaller than L for sufficiently large L. Then, to compute δρ(z, t f )δρ(z + u, t f ) one can perform a local density approximation and use the result of a homogeneous gas at a density n 0 (z). We then obtain

|ρ(q)| 2 = dz |ρ n0(z) (q)| 2 (C2)
where the subscript n 0 (z) specifies that one considers the result for a homogeneous gas of density n 0 (z). This expression is referred to as the local density approximation expression (LDA) of the power spectrum. We have tested this approximation, for conditions close to the experimental data presented in the main text, by comparing it with calculations based on the Bogoliubov excitations of the trapped system (see section E).

Appendix D: Time evolution of the density ripple power spectrum for a harmonically confined gas

Here we give an explicit derivation of Eq. ( 6) of the main text, for a gas harmonically confined in a longitudinal trap of frequency ω . Injecting Eq. ( 5) and Eq. ( 2) of the main text into Eq. (C2), and using the local initial power spectrum of θ which writes θ 2 q = mk B T /( 2 n 0 q 2 ), we derive Eq. ( 6) of the main text with

F = dzn 0 (z) sin 2 (c(z)qt) /N (D1)
where N is the total atom number. The density profile n 0 (z) is estimated itself within the LDA, using the local chemical potential

µ(z) = µ p (1 -(z/R TF ) 2 )), ( D2
)
where R TF is the Thomas-Fermi radius of the density profile and µ p is the chemical potential at the trap center. For a transverse harmonic confinement of frequency ω ⊥ , it has been checked, by comparing with predictions of the 3D Gross-Pitaevskii equation, that the equation of state of the gas is very well described by the heuristic formula [START_REF] Fuchs | Hydrodynamic modes of a one-dimensional trapped Bose gas[END_REF]

µ(n) = ω ⊥ √ 1 + 4na -1 , ( D3
) where a is the 3D scattering length between atoms. For small linear densities, we recover the 1D expression µ = 2 ω ⊥ an, valid far from the confinement-induced resonance [START_REF] Olshanii | Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons[END_REF]. Using Eq. (D3) and Eq. (D2), we obtain the density profile

n 0 (z) = η(1 -z2 ) + 1 2 -1 /(4a) (D4)
where we introduced z = z/R TF and η = µ p /( ω ⊥ ). This yields N = (4/3η + 8η 2 /15)R TF /(2a). The local speed of sound on the other hand, obtained from the thermodynamic relation c = n(∂µ/∂n)/m, writes

c(z) = c p (1 + η) (1 + η(1 -z2 )) 2 -1 (1 + η(1 -z2 )) ((1 + η) 2 -1) , ( D5
)
where c p is the speed of sound computed for the central density. Injecting into Eq. (D1), we then find

F = 1 4η/3 + 8η 2 /15 1 0 dz 1 + η(1 -z2 ) 2 -1 sin 2   τ (1 + η) (1 + η(1 -z2 )) 2 -1 (1 + η(1 -z2 )) ((1 + η) 2 -1)    . (D6)
When the gas is deeply 1D, namely for η 1, this expression reduces to

F 1D = 3 2 1 0 dz(1 -z2 ) sin 2 τ 1 -z2 . (D7)
Experimentally, values of η are in the range [0.6; 1.3]. Fig. 5 shows the function F, computed for η = 1. We compare it to F 1D and to the expression expected for a homogeneous gas, namely sin 2 (τ ). The Bogoliubov modes, indexed by an integer ν, then acquire an analytical dispersion relation and analytical wavefunctions that one can use for calculations. For each mode, the dynamics are accounted for by the harmonic oscillator Hamiltonian

H ν = ω ν x 2 ν 2 + p 2 ν 2 , ( E1
)
where ω ν = ω ν(ν + 1)/2 and x ν and p ν are canonically conjugate variables. The phase and density fluctu-ation operators write

θ(z) = ν θ ν (z)p ν δn(z) = ν n ν (z)x ν (E2) where      θ ν (z) = 1 √ 2 mg 2 np 1/4 √ 2ν+1 (ν(ν+1)) 1/4 P ν ( z R T F ) n ν (z) = √ 2ν+1 2RTF (ν(ν + 1)) 1/4 2 np mg 1/4 P ν ( z R T F ).
(E3) Here n p and R T F are the central density and radius of the Thomas-Fermi profile n 0 (z) = n p (1 -(z/R T F ) 2 ) and P ν are the Legendre polynomials. The interaction quench consists of a sudden change of the interaction parameter g from g i to g f = (1 + κ)g i at t = 0, while changing the longitudinal oscillation frequency by a factor √ 1 + κ so that R TF stays constant. Then the interaction quench preserves the shapes of the wavefunctions θ ν and n ν , and it simply changes the canonical variables x ν and p ν according to

x ν (t = 0 + ) = (g f /g i ) 1/4 x ν (t = 0 -) p ν (t = 0 + ) = (g i /g f ) 1/4 p ν (t = 0 -) (E4)
Under such a transformation, the initial thermal state, an isotropic Gaussian, becomes a squeezed state and its subsequent evolution under the Hamiltonian Eq. (E1) leads to a breathing of each quadrature. In particular

p 2 ν = p 2 ν i 1 + κ sin 2 (ω ν t) . (E5)
The initial value p 2 ν i is given by the thermal expectation value, which reduces to

p 2 ν i = k B T /( ω ν ) (E6)
for the low-lying modes for which k B T ω ν . Injecting Eq. (E2) into Eq. (A8), using Wick's theorem and the fact that different modes are uncorrelated we get

|ρ(q)| 2 = dαdX e iqX n 0 (α)n 0 (α + qt f )n 0 (α + X + qt f )n 0 (α + X) e -1 2 ν p 2 ν (θν (α)-θν (α+qt f )+θν (α+X+qt f )-θν (α+X)) 2 . ( E7
)
For qt f /m l c , where l c is the phase correlation length, one can expand the exponential and |ρ(q)| 2 is obtained by summing the contribution of each mode. Since the Legendre polynomials behave as cos((ν + 1/2)x + π/4) at small x, the contribution of the mode ν is peaked at q ν/R T F .

The predictions of Eq. (E7) may be compared to the one obtained within the Local density approximation. Here we focus on the case of thermal equilibrium. We compute the density ripple spectrum injecting the thermal equilibrium value Eq. (E6) and the mode wavefunc-tion Eq. (E3) into Eq. (E7). Fig. 6 shows the result for a cloud whose Thomas-Fermi radius fulfils l c /R TF = 0.2, where l c = 2 2 n p /(mk B T ) is the correlation length of the first order correlation function at the center of the cloud, and for a time-of-flight t f = 6 × 10 -4 mR 2 TF / . These parameters are close to the experimental ones. We compared the results with the LDA, together with the analytical formula for homogeneous gases Eq. (B3) and we find excellent agreement. We also compare with the LDA but using, instead of Eq. (B3), the approximation Eq. 5 of the main text. We find very good agreement as long The plot shows the density ripples spectrum of a gas at thermal equilibrium confined in a harmonic potential. The complete calculation, based on the expansion on the Bogoliubov modes, whose wavefunctions are given by the Legendre polynomial, is shown in solid line (blue). It is in excellent agreement with the spectrum computed within the local density approximation (LDA) shown in dashed line (red).

qR TF q ht f /m |ρ q | 2 /(R TF n p k B T t f /h) (
The further approximation of small wavevectors, Eq. ( 5) of the main text, injected into the LDA, shown in dotted line (green), is also in good agreement, for wavevectors fulfilling qRTF < 50. Calculations are done for a Thomas-Fermi radius lc/RTF = 0.2 and and time-of-flight t f = 0.015ml 2 c / , where lc = 2 2 np/(mkBT ) is the correlation length at the center of the cloud. These parameters are close to those of the experimental data.

as qR TF < 50.

Appendix F: Effect of a finite optical resolution and auto-correlation function

The effect of the imaging resolution is to multiply the theoretical power spectrum of density ripples with e -σ 2 q 2 , where σ is the rms width of the imaging pulse response function, assumed to be Gaussian. The resulting power spectrum, for a harmonically confined cloud at thermal equilibrium, is shown in Fig. [START_REF] Jaskula | Acoustic Analog to the Dynamical Casimir Effect in a Bose-Einstein Condensate[END_REF] for σ m/( t f ) = 0.85, a value typical for our experiments. The large q behavior of the power spectrum is highly dominated by the effect of resolution and only the first maximum of |ρ(q)| 2 remains visible. Fitting the experimental power spectrums for clouds at thermal equilibrium, we extract both the temperature and the imaging resolution (see Fig. (2) of the main text). The obtained rms widths σ, close to 3 µm, are compatible with the expected values if one takes into account the depth of focus of our imaging system ( 5µm) and the fact that, after the the expansion time t f the cloud explores several tens of µm along the imaging axis. Note finally that the imaging resolution is irrelevant for the investigation of

qR TF q ht f /m |ρ q | 2 /(R TF n p k B T t f /h) (b) 250 200 150 100 50 0 3 2.5 2 1.5 1 0.5 0 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 FIG. 7.
Effect of the finite resolution (color online). We consider a clould at thermal equilibrium in a harmonic potential with the same parameters as in Fig. 6. The power spectrum for infinite resolution (blue dashed curve) is compared to the power spectrum expected for a finite imaging resolution (red solid curve). The effect of the imaging resolution is to multiply the power spectrum with e -σ 2 q 2 , where σ is the rms width of the imaging pulse response function, assumed to be Gaussian. Here we took σ m/( t f ) = 0.85, a value close to that of experimental data.

the dynamics following an interaction quench, since, for each Fourier component q, we investigate the time behavior of the normalised quantity |ρ(q)| 2 (t)/ |ρ(q)| 2 i (see main text): the imaging resolution has no effect on this normalised quantity.

In our paper, we extract from the data the density ripple power spectrum since it is the relevant quantity that enable to resolve the collective Bogoliubov modes. Alternatively, one could consider the auto-correlation function of the density ripples C(u) = dz δn(z)δn(z+u) dz, which is the Fourier transform of the density ripple power spectrum: C(u) = 1/(2π) dq |ρ(q)| 2 e -iqu . In [START_REF] Manz | Two-point density correlations of quasicondensates in free expansion[END_REF], the authors introduced the normalised auto-correlation function g 2 (u) = 1 + C(u)/ du n(z) n(z + u) . Fig. [START_REF] De Nardis | Solution for an interaction quench in the Lieb-Liniger Bose gas[END_REF] shows g 2 (u) for the data at thermal equilibrium (before the quench) shown in Fig. (2) of the main text. A behavior very similar to that observed in [START_REF] Manz | Two-point density correlations of quasicondensates in free expansion[END_REF] is recovered.

Appendix G: Beyond instantaneous interaction switch off: finite transverse expansion time

In the data presented in the main text, the frequency of the probed longitudinal modes, of the order of cq, is no more than 0.15 × ω ⊥ . Then, due to the rapid transverse expansion, interactions during time-of-flight become almost instantaneously negligible and are expected to give only minor corrections to the density ripples spectrum computed for an instantaneous switching off of the interactions. It is nevertheless interesting to estimate their effect. This has already been computed in [START_REF] Hellweg | Phase fluctuations in BoseEinstein condensates[END_REF], in the limit µ ω ⊥ and using time-dependent Bogoliubov equations, i.e. equations of motion linearized in density fluctuations and phase gradient. The linearized calculations a priori require that density fluctuations stay small. Although in our case density ripples at the end of the time-of-flight have large amplitudes, the Bogoliubov calculations hold for the small q components, which fulfil q ml c /( t f ) and which are considered in our paper. The condition µ ω ⊥ on the other hand is not verified for the data shown in the main text. We nevertheless believe that the calculations of [START_REF] Hellweg | Phase fluctuations in BoseEinstein condensates[END_REF] give a relevant estimation of the effect of interactions during the time-offlight for our data. From results of [START_REF] Hellweg | Phase fluctuations in BoseEinstein condensates[END_REF], we find that the density ripples power spectrum for the small q wavevectors, given by equation ( 5) of the main text, should be corrected by the factor

C = (ω ⊥ t f ) -cq ω ⊥ 2 . (G1)
In all experimental situations C > 0.95, which confirm that the effect of interactions during the time-of-flight is small.

Appendix H: Effects which may reduce the oscillation amplitude

In this section we investigate two effects responsible for a reduction of the amplitude of the oscillations of J (see main text), as compared to the theoretical prediction given by Eq. ( 6) of the main text. We first consider the effect of the finite ramp time of the interaction strength, which reduces the squeezing of the Bogoliubov modes, as compared to an instantaneous quench. This effect contributes to the reduction of the amplitude on the order of 10%. We then investigate the reduction of the amplitude induced by the binning of the data with a finite resolution in τ . This effect amounts to an additional reduction of the amplitude by 18%.

1. Beyond the instantaneous quench: finite ramp time

In the experiment, the change of the effective interaction strength is not instantaneous: to ensure the adiabatic following of the transverse motion, we perform a ramp of the transverse oscillation frequency during a time t r . The finite value of t r is responsible for a decrease of the induced squeezing of each mode. In the asymptotic limit of very large t r , the squeezing vanishes since then, the modes follow adiabatically the modification of the interaction strength. In the following we compute the effect of the ramp on the squeezing of each mode and we use this result to compute the resulting decrease of the oscillation amplitude of J.

In order to estimate the effect of the finite ramp time, we will consider a homogeneous gas for simplicity. The Bogoliubov modes are then described by the Hamiltonian of Eq. ( 1) of the main text, namely

H q = A q n 2 q + B q θ 2 q . (H1)
We regard the effect of a ramp of ω ⊥ between the time t = 0 and the time t r : ω ⊥ goes from ω i ⊥ to ω f ⊥ = (κ + 1)ω i ⊥ , as depicted in Fig. [START_REF] Calabrese | Interaction quench in a LiebLiniger model and the KPZ equation with flat initial conditions[END_REF]. The coefficient B q = n 0 2 q 2 /(2m) is time-independent, while the coefficient A q evolves linearly during the ramp (i.e. during time interval 0 < t < t r ), since it is proportional to c 2 , itself proportional to ω ⊥ . Then, the solution of the second order equations describing the evolution of θ q and n q during the ramp is given in terms of the Airy functions. In order to investigate the squeezing, it is natural to introduce the reduced variables θq = θ q / θq ñq = n q /n q (H2)

where θq = (A q (t)/B q ) 1/4 and nq = (B q /A q (t)) where the matrix M has the following components:

       M 11 = (κ + 1) -1/4 π -B i (-δ -2/3 )A i (-(κ + 1)δ -2/3 ) + A i (-δ -2/3 )B i (-(κ + 1)δ -2/3 ) M 22 = (κ + 1) 1/4 π B i (-δ -2/3 )A i (-(κ + 1)δ -2/3 ) -A i (-δ -2/3 )B i (-δ -2/3 (κ + 1)) M 21 = (δ -4/3 (κ + 1)) 1/4 π -B i (-δ -2/3 )A i (-δ -2/3 (κ + 1)) + A i (-δ -2/3 )B i (-δ -2/3 (κ + 1)) M 12 = (δ -4/3 (κ + 1)) -1/4 π B i (-δ -2/3 )A i (-δ -2/3 (κ + 1)) -A i (-δ -2/3 )B i (-δ -2/3 (κ + 1)) (H4)
Here A i , B i are the first and second kind Airy functions and A i , B i , their derivatives and δ = κ/(t r ω i q ) the quench speed normalized to the initial mode frequency (we recall that the quench strength is κ = ω f ⊥ /ω i ⊥ -1). Under this transformation, the initial isotropic Gaussian distribution transforms into a squeezed distribution, i.e. a Gaussian elliptical distribution with a squeezing angle α and ratio between the rms width of the two eigenaxes equal to the squeezing factor S. In order to find α and S, let us compute, for any angle β, the width along the quadrature xβ = cos(β) θq + sin(β)ñ q . Using the fact that the initial state is a thermal equilibrium state fulfilling θ2 q i = ñ2 q i ≡ V and θq ñq i = 0, and using the transformation above, we find

x2 β = V {cos 2 (β) M 2 11 + M 2 22 +sin 2 (β) M 2 21 + M 2 22 + 2 cos(α) sin(α) (M 11 M 21 + M 22 M 12 )}. (H5)
The squeezing angle α is found by imposing 9) for quench amplitudes κ = 2 and κ = 4 as a function of ω f q t r where ω f q is the final frequency of the mode. For very slow modes ω f q t r 1, one recovers the results expected for an instantaneous quench : α 0 and (S 2 -1) κ. For modes of larger frequency, the effect of the ramp is to reduce the squeezing and also to rotate its axis.

d x2 β /dβ β=α = 0, which leads to tan(2α) = -2 M 11 M 21 + M 22 M 12 M 2 21 + M 2 22 -M 2 11 -M 2 12 . ( H6
The post-quench dynamics results in a breathing of the θq quadrature: θ2 q oscillates with an amplitude V (S 2 -1)/S. Coming back to the variable θ q , the evolution at times t > t r writes

θ 2 q (t) = θ 2 q i √ κ + 1 S q 1 + (S 2 q -1) sin 2 (ω f q (t -t r ) + α q ) (
H7) where the indice q in S and α indicates these quantities depend on q. As seen in Fig. [START_REF] Calabrese | Interaction quench in a LiebLiniger model and the KPZ equation with flat initial conditions[END_REF], the angle α q is very close to ω f q t r /2, for moderate values of ω f q t r . Injecting this value into Eq. (H7), we find that it amounts to shifting the time reference to t r /2. We perform this shift when ⊥ /ω i ⊥ -1 = 2 (solid lines) and κ = 4 (dashed lines). (c) shows √ κ + 1(S 2 -1)/(Sκ), which gives the amplitude of the resulting breathing oscillations normalized to the amplitude for an instantaneous quench (see text), versus ω f q tr where ω f q is the final frequency of the mode. The squeezing angle is shown in (d), normalized by ω f q tr/2.

(a) 0 t r ω f ⊥ ω i ⊥ t ω ⊥ κω i ⊥ ñq α σ 1 σ 2 σ 1 /σ 2 = S θq (b) ω f q t r α/(ω f q t r /2) (d) 1.2 0.8 0.4 0 1.1 1.05 1 0.95 0.9 ω f q t r (S 2 -1)
√ κ + 1/(Sκ) (c
analyzing the data, in other terms the reduced variable τ is τ = cq(tt r /2).

Let us now consider the evolution of the density-ripples power spectrum |ρ q | 2 (t). For small q, |ρ q | 2 (t) is proportional to θ q | 2 (t) such that the evolution of |ρ q | 2 (t) is given by Eq. (H7). This leads to, J(q, τ ) = √ κ + 1 S q 1 + (S 2 q -1) sin 2 (τ ) .

(H8)

Let us now investigate the quantity J(τ ), defined in the 10. Effect of the finite ramp time of the interaction strength for a homogeneous gas. The expected behavior of J (solid line) is compared to the case of an instantaneous ramp (dashed line). Here we consider a gas of Rubidium atoms at conditions close to the experimental ones. More precisely, the linear density is n0 = 630 atoms per µm, the initial transverse oscillation frequency is ω ⊥ = 2π × 1.5 kHz, the quench strength is κ = 2 and the ramp time is tr = 0.7 ms. The range of q values used to compute J is q ∈ [0.1, 0.5]µm -1 and the range of measurement times is t ∈ [tr/2, 6ms]. main text for experimental data. Here we will assume that the measurement times are spread over [t m , t M ] and we denote h(t)dt the number of points in the time interval [t, t + dt]. The q values are assumed to be equally spaced, as in the case of a Fast Fourier Transform, and only q values in the interval [q m , q M ] are considered. We assume that J(τ ) is obtained by binning in τ the collection of data with a bin size ∆ small enough so that, for all measurement times t, J(q, τ ) is about constant in the interval q ∈ [τ /(ct), (τ + ∆)/(ct)]. Then, one has J(τ ) = 1 h(t)dt∆/(ct) h(t)dtJ(q = τ /(ct), τ )∆/(ct), (H9) where the integrals are evaluated between t 1 and t 2 , where t 1 = Max(t m , τ /(cq M )) and t 2 = Min(t M , τ /(cq m )). Typically, in the experiment small times are sampled more densely than large times. Taking h proportional to 1/t, we obtain J(τ ) = 1 dt/t 2 dt J(q = τ /(ct), τ ) t 2 = dqJ(q, τ ) q 2 (τ )q 1 (τ ) , (H10) where q 1 = max(τ /(c)t M , q m ) and q 2 = min(τ /(c)t m , q M ).

The predicted time evolution of J is shown in Fig. [START_REF] Cazalilla | Quantum quenches in the Luttinger model and its close relatives[END_REF] for parameters close to that of the experimental data shown in the main text. The amplitude of the first oscillation is decreased by about 10%.

Finite width of the convolution function used in data processing

The data shown in the inset of Fig.

(3) of the main text correspond to a data set with an exceptionally good signal over noise. In general, the spread of the data points corresponding to a given value of τ (and thus corresponding to different times t and wavevectors q) is as large as about 50%. In such conditions, a binning of the data as a function of the reduced time τ = cqt with a bin size sufficiently large to accommodate many data points is required in order to increase the signal over noise. As describe in the main text, we use a "smooth" binning: we compute the weighted average of the data, J, with a Gaussian cost function of rms width ∆. For a very dense data set, we can define the local average value J(τ ) = i,τi∈[τ,τ +dτ ] J i /dτ , where the sum is done on the data set and dτ is much smaller than ∆. Then J corresponds to the convolution of J with a convolution width ∆. This convolution reduces the amplitude of the oscillations. To estimate this amplitude reduction, let us disregard the small damping of the oscillations coming from the cloud inhomogeneity (see section 3) and thus consider data which would follow the oscillatory behavior J = A sin 2 (τ ). The smoothing J(τ ) = ∞ -∞ dτ J(τ ) e -(τ -τ ) 2 /(2∆ 2 ) /( √ 2π∆ 2 ) reduces the amplitude to A = Ae -2∆ 2 . For ∆ = 0.1π, as used for the data analysis shown in the main text, the amplitude is reduced by 18%.

FIG. 2 .

 2 FIG. 2. Density ripples analysis (color online). (a)Typical absorption image (optical density shown) taken after a timeof-flight t f = 8 ms. (b) Power spectrum of density ripples, obtained by averaging over about 50 images, for a cloud at thermal equilibrium containing 16000 atoms confined in a trap with frequencies ωz/(2π) = 8.5 Hz and ω ⊥ /(2π) = 1.5 kHz, yielding a Thomas-Fermi radius RT F = 75 µm. The dashed (green) line is a theoretical fit (see text), yielding a temperature T = 55 nK and an optical resolution σ = 2.9 µm. (c) Power spectra after a quench of strength κ = 2, at times t = 2.1 ms (crosses, green), t = 2.6 ms (circles, blue) and t = 4.6 ms (squares, black), the solid (red) curve being the initial power spectrum.

2 JFIG. 3 .

 23 FIG.3. Time evolution of squeezed collective modes produced by an interaction quench. The normalized density ripples power spectrum is plotted versus the reduced time τ = cqt, where the speed of sound c is calculated for the central density. Inset shows the data corresponding to each measurement time and discrete q values, for a data set corresponding to κ = 2 and ω ⊥,i = 2π × 1.5 kHz, together with the resulting continuous averaged quantity J (see text). Orange crosses correspond to t < t g 1 th and blue circles to t > t g 1 th . The main graph shows the evolution of the experimental smoothed quantity J for different data sets. The error bars show the typical statistical uncertainty on J. The initial transverse oscillation frequency is 1.5 kHz, except for the thick dark grey (blue) curve for which it is 3 kHz. Quench strengths are κ = 4 (light gray (orange)), κ = 2 (dark gray (blue)) data and κ = -0.7 (black). Dashed lines are theoretical predictions for quench strengths κ = 2 (lightgray (orange)), 1 (light gray) and -0.35 (black).

FIG. 5 .

 5 FIG.5. Oscillation of each spectral component of the power spectrum for a harmonically confined gas in the LDA (color online). The function F is shown in thick solid lines (green), for η = µp/( ω ⊥ ) = 1.0. The pure 1D limit, corresponding to η 1 is shown as dashed (red) lines. The undamped oscillations expected for a homogeneous gas are shown in dotted (blue) line. In all the cases, τ = cqt where c is the central sound velocity.

FIG. 8 .

 8 FIG. 8. Normalised auto-correlation function of the density ripples. The data set used is the same as that of Fig. (2)(b) of the main text. Experimental data are shown in green and the theoretical prediction for a cloud at a temperature T = 55 nK and an optical resolution σ = 2.9 µm is shown in blue.

)

  The most squeezed quadrature is xα while xα+π/2 is the most anti-squeezed quadrature. The squeezing factor is S = x2 α / x2 α+π/2 . It also writes S = x2 α /V since the conservation of the phase-space area ensures x2 α x2 α+π/2 = V , and it is evaluated injecting β = α in Eq. (H5). Results are shown in Fig. (

8 FIG. 9 .

 89 FIG.9. Effect of the interaction strength ramp on the squeezing of longitudinal modes. The time sequence is shown in (a). An example of the phase space distribution at the end of the is shown in (b): the 1/ √ e line of the Gaussian distribution is plotted. The squeezing factor S is the ratio between the rms widths along the anti-squeezed and the squeezed directions. The curved arrow shows the direction of rotation under free evolution. Quantitative results are shown in (c) and (d) for a quench strength κ = ω f ⊥ /ω i ⊥ -1 = 2 (solid lines) and κ = 4 (dashed lines). (c) shows √ κ + 1(S 2 -1)/(Sκ), which gives the amplitude of the resulting breathing oscillations normalized to the amplitude for an instantaneous quench (see text), versus ω f q tr where ω f q is the final frequency of the mode. The squeezing angle is shown in (d), normalized by ω f q tr/2.

  FIG.10. Effect of the finite ramp time of the interaction strength for a homogeneous gas. The expected behavior of J (solid line) is compared to the case of an instantaneous ramp (dashed line). Here we consider a gas of Rubidium atoms at conditions close to the experimental ones. More precisely, the linear density is n0 = 630 atoms per µm, the initial transverse oscillation frequency is ω ⊥ = 2π × 1.5 kHz, the quench strength is κ = 2 and the ramp time is tr = 0.7 ms. The range of q values used to compute J is q ∈ [0.1, 0.5]µm -1 and the range of measurement times is t ∈ [tr/2, 6ms].

ACKNOWLEDGMENTS

This work was supported by Région Île de France (DIM NanoK, Atocirc project). The authors thank Dr Sophie Bouchoule of C2N (centre nanosciences et nanotechnologies, CNRS / UPSUD, Marcoussis, France) for the development and microfabrication of the atom chip. Alan Durnez and Abdelmounaim Harouri of C2N are acknowledged for their technical support. C2N laboratory is a member of RENATECH, the French national network of large facilities for micronanotechnology. M. Schemmer acknowledges support by the Studienstiftung des Deutschen Volkes.

APPENDIX

This appendix gives technical information and details of calculations. In Appendix A we give a general derivation of the density ripples power spectrum, which does not a priori assume a homogeneous system. Appendix B gives the result for a homogeneous system and the analytical prediction for thermal equilibrium [37]. Appendix C details the derivation of the density ripple power spectrum for a trapped gas, computed using the results for homogeneous gases and the local density approximation.

Appendix D provides the explicit calculation of the post-quench evolution of the power spectrum for a harmonically trapped gas, namely the calculation of the function F of the main text. In Appendix E we verify the validity of the local density approximation for the parameters of the data presented in the main text. For this purpose, we compute the density ripple power spectrum using the Bogoliubov modes of the trapped gas. In Appendix F, we investigate the effect of finite resolution on the measured density ripple power spectrum. We also make the link between the power spectrum and the auto-correlation function, which permits to compare our data at thermal equilibrium with previously published work. In Appendix G, we justify that interactions play a negligible role during time-of-flight, so that the calculations of the density ripples power spectrum, which assume instantaneous switch-off of the interactions, are valid. In Appendix H, we investigate two effects responsible for a reduction of the oscillation amplitude of the quantity J(τ ), extracted from the data, as compared to the simple theoretical predictions Eq. ( 6) of the main text: First the finite ramp time of the interaction strength decreases the squeezing of the collective modes, and second the finite resolution in τ resulting from data binning is responsible for a decrease of the expected oscillation amplitude on the processed data.

Appendix E: Beyond the LDA: calculation using Bogoliubov modes of a harmonically confined 1D gas Here we consider a 1D gas confined longitudinally in a harmonic trap of frequency ω . In opposition to the calculations done in the previous section we do not rely on the local density approximation but use the Bogoliubov modes of the trapped gas to compute the post-quench evolution and the density ripples power spectrum. The relevant collective modes lie deep in the phononic regime.


Y Bai

G Fraisse

F Wurtz

A Foggia

Y Deless

F Domain

Experimental and numerical study of a directly PV-assisted domestic hot water system

Keywords: Domestic hot water, photovoltaic, circulation pump, pump start-up, experimentation, simulation

. However, the control mode of conventional SDHWs induces additional costs related to the consumption of auxiliaries and other risks of dysfunction of the circulation pump due to the temperature probes and controller setup which induces a lower annual productivity of solar collector (200 instead of 400 kWh/m 2 ). From this point of view, the photovoltaic pumped system seems suitable since it eliminates the controller and temperature sensors. This paper focuses on an experimental and numerical study of the behavior of a PV-SDHW system, focusing on the start-up phase optimized through various electronic devices. A detailed model of a circulation pump was developed by considering a direct current (DC) circulation pump coupled with various electronic devices (linear current booster and maximum power point tracker). The developed models were then validated experimentally, to reveal the influence of the threshold solar radiation on the circulation pump start-up and the pump flow rate as a function of the solar radiation, and its effects on the annual energy performance of PV-SDHW systems.

Introduction

Reducing the greenhouse gas emissions is essential for limiting climate change. According to the report (Seghier, 2006), based on data from 41 industrialized countries, 36.8% decrease in emissions in the countries of Central and Eastern Europe, however, emissions decreased slightly between 1990and 2001. Still, the 2000-2001 periods showed an increase despite the commitments of the Kyoto Protocol. At the European level, the "triple 20" goal for 2020 aims to reduce greenhouse gas emissions and energy consumption by 20%, respectively and simultaneously incorporate 20% renewable energy into energy consumption.

In France, emission from the building sector accounts for 25% of the total emission, producing 120 million tons of CO 2 . Heating and domestic hot water (DHW) currently take the largest share of energy consumption in buildings (72% for heating and 11% for DHW). Thermal regulation has greatly reduced energy consumption for heating, which enlarges the share of DHW. Under these conditions, the role of SDHW is increasingly important in terms of constructing buildings with very low energy consumption.

A SDHW system reduces energy consumption and covers the demand of domestic hot water by 30-70%.

However, the results of monitoring 120 SDHW systems in France (Buscarlet et al., 2006) showed the dysfunction problems due to system over-sizing (cold water temperatures overestimated by about 4 -5 °C and the hot water draw profile), the incorrect setting of the controller, and temperature probes that are defective or not properly installed , as also highlighted in another report [START_REF] Buscarlet | Plan soleil, catalogue des défauts des installations de chauffe-eau solaires individuels[END_REF], the latter problem could lead a disparity of the pump working time as well as a reduced system energy gain.

To minimize the shortages identified in the current installations, the photovoltaic (PV) -assisted operation is a potential measure. The first alternative approach is to replace the thermal solar collector in the conventional SDHW systems with PV modules. In this new system, hydraulic piping and heat exchangers are no longer used, which could reduce heat loss and improve heat transfer efficiency. [START_REF] Ghoneim | Economic analysis of photovoltaic-powered solar domestic hot water systems in Kuwait[END_REF] showed that this type of system can suffice for DHW needs, while the overall system cost is close to a conventional fuel based system. With the reduction in the cost of PV modules, the PV-SDHW system will be more competitive in the near future.

Another PV-assisted application is to couple the circulation pump directly with a PV module. The necessary PV module output power for this type of PV-SDHW is much lower than the previous configuration. The controller and the temperature sensors are not necessary in this system. Several studies have been conducted aiming to evaluate energy and economic performance through experimental and/or simulation results. Numerically, most of the studies are based on the use of very simplified circulation pump models. Other studies have improved performance by optimizing the system components or operating conditions.

Dayan [START_REF] Dayan | High performance in low-flow solar domestic hot water systems[END_REF] studied different SDHW configurations, specially for the PV-assisted SDHW system. The simulation was performed with a flat-plat solar thermal collector of 3.185 m 2 , a tank of 400-l and an average daily hot water draw is about 300-l (12.2 kg/hr). An indirect natural convection using an external plate heat exchanger is considered in this study. The circulation pump in the solar loop is powered by a 20-Wp PV module.

A pump model was developed, and the flow is determined from a third order polynomial function based on the voltage V and hydraulic head H. This model was also used in others studies of PV pumping systems. [START_REF] Kou | A method for estimating the long-term performance of direct-coupled PV pumping systems[END_REF][START_REF] Cardinale | Economic optimization of low-flow solar domestic hot water plants[END_REF][START_REF] Martire´ | A simplified but accurate prevision method for along the sun PV pumping systems[END_REF]). The annual performance was analyzed numerically by estimating the solar fraction. The author concluded that the solar fraction obtained with the PV-SDHW system is lower compared with a standard SDHW system. Meanwhile, the low flow rate of 15 l/h.m² compared with standard flow rate (50-70 l/h.m²), can maintain the stratification in the tank and therefore augment the gain of solar energy by 8 % (Castell et al, 2010;Fanney and Klein, 1998;[START_REF] Kenjo | Etude du comportement thermique d'un chauffe-eau solaire à faible débit[END_REF][START_REF] Knudsen | Thermal stratification in vertical mantle heat exchangers with application to solar domestic hot water systems[END_REF]).

However, the circulation pump consumption is lower with PV power and the corrected solar fraction, which considers that the parasite energy used by the pump and the controller are very similar. [START_REF] Al-Ibrahim | Design Procedure For Selecting An Optimum Photovoltaic Pumping System In a Solar Domestic Hot Water System[END_REF] also conducted a numerical study on the optimization of a PV-SDHW system. The optimization procedure is divided into two phases. Initially, it aims to find the flow profile in relation to solar radiation, which maximizes the solar fraction. In the second phase, the goal is to select components from a range of pumps supplied by PV modules in order to obtain the pump torque/speed profile whose flow rate versus solar radiation profile approach the optimum. With this optimized approach, they found that the direct system (without heat exchanger) can achieve a solar fraction of about 0.7 versus 0.635 for an indirect system with a heat exchanger and a natural convection regime and 0.504 for an indirect system with a heat exchanger in a forced convection regime. Here, only the solar fraction is used as an estimation factor and other evaluation criteria (economic and environmental) were not considered. Finally, no experimental results are available for validating the models. [START_REF] Cardinale | Economic optimization of low-flow solar domestic hot water plants[END_REF]) conducted an experimental and numerical study of a PV-SDHW system.

The system studied produces 500-l of hot water per day for a family of four. An external heat exchanger was used to maintain the stratification in the storage tank with low flow. Two circulation pumps are powered directly by PV modules, one for the solar-heat exchanger loop and the other for the tank-exchanger loop. The system's performance is evaluated numerically using the TRNSYS simulation tool (Klein et al., 1998). A pump model has been developed with the flow also determined by the pump hydraulic head and voltage. The difference between this analysis and Dayan's is that in the latter, the cycle analysis method (life cycle analysis, LCA) was taken into account. They compared several configurations of PV-SDHW by varying the collector surface, the exchanger surface, and the weather conditions. They concluded that the best performance is achieved with the maximum thermal collector area and the maximum heat exchanger surface. However, the model does not take into account pump efficiency. They consider that the power supplied to the motor is fully transferred to the pump. In fact, for small power motors, motor losses are not insignificant and efficiency is often low ranging 15 -40%.

Macleod [START_REF] Macleod | Evaluation of components in solar water heaters with photovoltaic powered pumps[END_REF] conducted a study on a PV-SDHW system using a 'solar boiler' provided by the manufacturer, Thermo Dynamics Limited. The system mainly includes two 6-m² flat plate solar thermal collectors and a 450-l storage tank with an internal coil heat exchanger. The subsystem consists of two polycrystalline PV modules (17 W and 20 W), two DC motors, and centrifugal pumps manufactured by Procon.

A linear current booster (LCB) is placed parallel between the PV module and the motor to ensure that the pump starts in low solar radiation. The different configurations depending on the power of the PV module and motor were analyzed experimentally. From the experimental results, the author found that the LCB can provide sufficient power to start the motor under the low solar radiation level, but during start-up, the performance of the LCB (ratio of power supplied to the motor and power output by the PV module) is low, between 65 and 70%.

For configurations using a high-power motor, the start takes place earlier with solar radiation at approximately 250 -300 W/m². The performance of the LCB is slightly higher (1-2%) than the other configurations.

Grassie et al. [START_REF] Grassie | Design of a PV driven low flow solar domestic hot water system and modeling of the system collector outlet temperature[END_REF]) also conducted a study on a system with the PV-SDHW system aiming to provide a constant temperature at the collector output and to maintain thermal stratification in the tank. A numerical model was developed to determine the temperature at the collector output versus the pump flow rate, the collector area, the solar radiation, the temperature at the collector inlet, and the desired temperature at the collector outlet. To maintain this temperature, the power output of PV modules is modulated by varying the number of cells in series and thus the flow rate through the solar thermal collector. The results show that the predetermined temperature of the collector output is obtained for a sunshine level between 500 and 1000 W/m² when 66% of the PV module area is hidden. However, this model could be improved by taking into account the effects of the incidence angle and the viscosity of the fluid.

More concretely on the selection of the motor and pump types for directly coupled PV-assisted SDHW system, the use of a brushless DC motor is found to be more suitable which presents a high efficiency, high reliability and minimum maintenance requirement (Langridge et al., 1996;[START_REF] Kou | A method for estimating the long-term performance of direct-coupled PV pumping systems[END_REF]Firatouglu and Yesilata, 2004;). However, Brushless DC motors are available for low power applications. Concerning the pumps driven by DC motors, most of the studies indicated that the brushless DC motor coupled to a centrifugal pump was the most suitable combination (Anis, W.R., [START_REF] Metwally | Dynamic performance of a directly coupled PV pumping system[END_REF][START_REF] Al-Ibrahim | Design Procedure For Selecting An Optimum Photovoltaic Pumping System In a Solar Domestic Hot Water System[END_REF]Firatouglu and Yesilata, 2004;) Most studies are based on the use of simplified models and experimental results which can not determine the level of annual performance on this type of PV-SDHW systems compared to a conventional system. This paper therefore focuses on the experimental and numerical studies of the solar circulation pump component supported by a PV module. The model was then used to evaluate the annual performance of the overall system.

The PV-SDHW system

The experimental set-up

An experimental set-up (Fig. 1) was built at the French National Institute for Solar Energy (INES). It comprises two flat plate solar thermal collectors (2 and 4 m², Fig. 2) connected in parallel. The 400-l storage tank is equipped an internal coil heat exchanger (Fig. 3). With the NAPAC acquisition system (which also controls the system) and the replacement of the circulation pumps, the standard SDHW systems and different configurations of the PV-SDHW system can be studied. The resistances R, R 1 , R 2 are used to determine the current in the circuits by measuring the voltage across the resistances.

Experimental results and discussions

The experimental results have allowed us to study the electrical, hydraulic, and thermal behaviors of different configurations.

Electrical component

The electrical behavior can be described in three steps during the circulation pump's start-up phase. In step 1, the voltage and current increase proportionally with the increase of solar radiation. In step 2, when sufficient voltage and current are supplied to the pump, the integrated control system starts to work. The current and voltage vary alternately, gradually increasing over time. The duration of this period depends on the intensity of the solar radiation. The motor begins to rotate in the third stage. However, at the beginning of this period, the pump flow rate is too low to be detected. After the start-up, the current varies in proportion to the solar radiation. This motor start-up process is also highlighted theoretical in the study realized by Suehrcke et al. (Suehrche et al., 1997). Fig. 5 shows the variation of the voltage and current for the solar circulation pump on a sunny day. With the MPPT, the pump start-up occurred earlier, at a solar radiation level of about 200 W/m² (this value varies up to 250 W/m² depending on the day studied) versus 350 W/m² for the standard DC circulation pump (config. I). The current in the pump varies proportionally with the solar radiation level, while the voltage remains almost constant between 14 and 16 V. This value is directly related to the operation of the MPPT, which can deliver greater power to the solar circulation pump than that to the standard DC circulation pump with the same level of solar radiation. When the pump rotation speed is high enough to generate enough electromotive force, the voltage across the pump is close to the LCB reference voltage. The LCB stops working when the voltage of the pump exceeds the LCB reference voltage. In the experimental configuration, the voltage across the pump is always lower than the LCB reference voltage and it remains at about 10 V. The performance of the LCB (defined as the ratio of the power supplied to the circulation pump to the power delivered by the PV module) is relatively low during the pump start-up (60%) and stopping. After the start-up, the efficiency reached 95%. It decreases again when the pump stops working.

Hydraulic and thermal behaviors

This section presents the experimental results concerning the profile of the circulation pump flow rate versus solar radiation, the threshold of solar radiation for pump start-up, and the temperature changes across the thermal collector.

For all configurations, the pump flow rate varies almost proportionally to solar radiation, as shown in Fig. 8. The maximum flow rate in the solar circulation pump (config. II) is nearly the same as that in the standard DC pump with a LCB (config. III), except about 15 l/h.m² higher than that in the standard DC pump (config. I). It reduces the outlet temperature of the thermal collector. The heat losses are reduced and the annual energy performance could be augmented. The pump flow rate in the afternoon is much higher than in the morning with the same level of solar radiation.

The difference is close to 10 l/(h.m²). Most studies of PV-SDHW systems such as the numerical studies of Al-Ibrahim and Kou (Al-Ibrahim, [START_REF] Kou | A method for estimating the long-term performance of direct-coupled PV pumping systems[END_REF] do not take it into account for this important phenomenon.

The flow affects circulation pump efficiency in terms of overall losses (friction losses in the hydraulic pipe lines) and fluid characteristics (temperature, density, viscosity, etc.). According to several studies conducted on the characteristics of circulation pump [START_REF] Henriksson | Improving pump efficiency and system performance by selecting the optimum fluid viscosity grade[END_REF][START_REF] Alibert | Efficiency advantages in Vane, Piston and Gear Pumps-What high VI Hudraulic Fluids Can Do for you[END_REF], viscosity is a major factor affecting on the performance of the circulation pump whose value is quite low (14% maximum). The viscosity of the coolant can be calculated depending on the temperature and the percentage of anti-freeze [START_REF] Guergen | Automatique des chauffe-eau solaire individuels grâce à l'alimentation photovolataique du circulateur[END_REF].

( )

1.42 % 309 ( 37) 1.1 256 0.0001 100

cor fluid fluid antifreeze T Exp T υ -   = ⋅ + ⋅ ⋅ - ⋅     (1)
With:

υ cor , kinematic viscosity, [mm²/s] T fluid , fluid temperature, [°C]
The viscosity of the fluid decreases with increasing temperature, which results in less mechanical loss in the pump. The higher the viscosity, the lower the performance of the pump. This influence is particularly marked for centrifugal pumps. A factor related to the viscosity can be used (engineering toolbox) (Nally, 2007) to take into account this effect to calculate the pump efficiency.

The temperature difference between the collector's input and output is high for both the standard DC circulation pump (config.I and III) and the solar pump (config.II) (15-20 °C), as illustrated in Fig. 10. This discrepancy is much greater than the temperature differential in the classic matched flow rate system (∆T=5 °C). This does not produce favorable results in terms of the heat loss of the thermal collectors.

Figure 9: Comparison of the temperature difference between the inlet and outlet of the solar thermal collector versus solar radiation

Circulation pump dynamic model associated with electronic devices

The experimental results have highlighted the need to develop a detailed model of the circulation pump if the goal is to accurately evaluate energy performance by numerical simulation. We therefore first developed the model of the standard DC circulation pump consisting of a brushless DC motor and a centrifugal pump. The model of the solar circulation pump was then obtained by integrating the MPPT model.

Brushless motor model

Regarding the electrical, electronic, and electromagnetic modeling of the brushless motor, the following hypotheses were taken into account:

-The magnetic circuit is not saturated, a condition necessary to consider the fluxes as linear functions of current;

-The magnetic circuit is fully laminated, so that only the windings (inducing and induced) are driven by currents. In addition, the current density can be considered uniform in the section of a single conductor;

-The consumption of the motor integrated with the control device is estimated using a resistor R c connected in series with the rotor.

The electrical diagram of the motor is shown in Fig. 10. E m , motor electromotive force, a function of the motor rotation speed,

m V E K ω = [V] (3) With K V , voltage constant, [V.s.rad -1 ]
The motor torque T m is proportional to the motor armature current I a ,

m m a T K I = [Nm] (4)
The mechanical equation including the rotation speed is written as:

1 1 m L d T T A B J dt ω ω = + + + [Nm] (5)
Where The pump load torque can be expressed by the following equation:

1.8 2 2 L T A B ω = + [Nm] (6) With A 2 , torque constant of centrifugal pump, [Nm] B 2 , torque constant of centrifugal pump, [Nm.rad -1 .s]
By combining equations ( 5) and ( 6), the motor torque can also be expressed as:

1.8 1 1 2 2 m d T A B A B J dt ω ω ω = + + + + [Nm] (7)
The coefficients K v and K m are close and it is possible to consider them as equal by ignoring the losses in the electromechanical power conversion.

The definition of the model takes into account the three steps identified during the test (Fig. 11). The equations above are adapted to the three steps of the circulation pump operation.

During

Step 1, the electronic control device in the motor starts to run. The variation of current and voltage increases linearly as solar radiation increases. The changes in voltage and current obey the Ohm law. During this period, the tension and intensity levels are too low for the rotor to rotate. At the end of step 1, tension and intensity are sufficient to start the electronic control systems.

When sufficient voltage and current are supplied to the circulation pump, the electronic control device promoting the pump start-up begins to operate.

Step 2 corresponds to the transient regime: voltage and current vary alternately. The pump rotation speed is zero as well as the load torque (T L = 0), Equation (2) becomes:

( )

a a a a m R I L dI dt V + = (8)
After starting (step 3), the derivative of the current is insignificant because E m >> ( )

a a
L dI dt . It is therefore assumed that this period corresponds to a steady state.

Equation ( 2) thus becomes:

m a a m E R I V + = (9)
By combining equations ( 2) and ( 9), we obtain:

V a a m K R I V ω + = (10)
Equation ( 11) concerning the motor torque is written as:

1.8 1 1 2 2 m a K I A B A B ω ω = + + + (11)
Parameters R a (resistance of the armature), L a (inductance), and K v (voltage constant) are obtained experimentally.

Models of electronic devices

In the experimental section, we showed that electronic devices such as the MPPT and LCB can improve the performance of the circulation pump toward the start-up. Thus, the models concerning these electronic devices were also developed.

MPPT model

The essential parameter for determining the output of the MPPT is the transformation ratio, which can be determined by the following relationship [START_REF] Appelbaum | The operation of permanent magnet DC motors powered by a common source of solar cells[END_REF]:

1 2 2 2 2 ppm a ppm MPPT ppm MPPT ppm MPPT I R E E n V V V η η η       = + +           (12)
Where, V PPM and I PPM are the voltage and current at maximum power point, respectively. E m is the motor electromotive force without MPPT and the η MPPT represents MPPT efficiency.

Here we assume that the efficiency of the MPPT is equal to 0.5 for solar radiation, which is less than 200 W/m² and 0.9 for sunlight above this threshold because MPPT performance is not stable at low levels of solar radiation.

The motor voltage and current are related to V m and I m by:

m MPPT ppm V nV η = (13) m ppm I I n = (14)

Model of linear current booster

There are few numerical studies on LCB, and thus a model based on experimental results has been developed.

The voltage across the PV module remains constant when LCB works; the value is equal to the LCB reference voltage.

int

PV setpo V V = ( 15
)
LCB efficiency is determined as the ratio between the power output of the PV module and the power delivered to the circulation pump.

LCB m m PV PV I V I V η = (16)
A transformation ratio is used for LCB:

PV m nI I = ( 17
)
This ratio is determined as a function of the solar radiation in the present study:

1 2 n l G l = + (18)
Where l 1 and l 2 are the coefficients obtained experimentally.

The pump voltage is written as:

m LCB PV V V n η = (19)

Centrifugal pump model

The model of the centrifugal pump is presented in Dayan [START_REF] Dayan | High performance in low-flow solar domestic hot water systems[END_REF]. The pump flow rate is calculated according to the voltage V across the circulation pump and the hydraulic head H:

2 2 2 2 3 2 3 2 m m m m m m m V kH V jH iHV hHV gV fV eV dH cH bH a m + + + + + + + + + + = ⋅ [kg.s -1 ] (20)
The coefficients used in equation ( 20) are determined experimentally.

Model validation

The models developed have been implemented in the TRNSYS environment [START_REF] Klein | TRNSYS[END_REF] in order to simulate a complete PV-assisted SDHW system. Firstly, it is important to validate the models before establishing a complete system. A comparison is presented in the following paragraph between the simulation results of the circulation pump model and the experimental results with different configurations.

The simulation results are compared with experimental data during a sunny day for the solar circulation pump.

Fig. 12 and Fig. 13 show the voltage and current of the solar circulation pump. The difference between the calculated and the measured current is greater, as shown in Fig. 12. In the morning, between t = 12,000 s and t = 16,000 s, the relative error varied between 3% and -10%. The numerically calculated voltage shows the same trend but the errors are smaller (± 5%). experimental data, the greatest difference is found in the early morning (t = 0 s to t = 40,000 s) because of the unstable operation of the circulation pump as it starts up. In addition, under the low solar radiation condition, the MPPT performance is low and varies much more. This gap tends to decrease when the sun rises. ( )

1 2 col colsimu cor Q Q C C υ = + [l/h] (21)
With C 1 and C 2 the coefficients related to the viscosity, obtained experimentally. The effect of viscosity on the pump flow rate is correctly considered, as shown in Fig. 15. presents the lowest consumption of the auxiliary heater. The PV-SDHW system using a standard DC circulation pump, which consumes slightly more than 2,000 kWh annually, is the least efficient. The consumption of the PV-SDHW system with the solar circulation pump is between the matched flow rate systems with ∆T=9 °C and with ∆T=13 °C. This is logical because the temperature difference between the inlet and outlet of the collector for the solar pump is about 11 °C (Fig. 9). However, with the PV-SDHW, the circulation pump uses the electricity generated by the PV module and there is no controller. The total consumption of the system with the solar circulation pump is therefore lower than the matched flow rate system, even for the most efficient consumption (∆T = 5 °C). The system with a LCB also shows a low total consumption compared to the on-off system, but it is not the best solution. The solar fraction (ratio of energy supplied by the solar part at the heat exchanger and the total system load energy), the most favorable is obtained with the matched flow rate system with ∆T=5 °C (0.61). When the temperature difference increases, the solar fraction decreases. The solar fraction for the PV-SDHW system with a standard DC circulation pump is the lowest since it starts relatively late compared to the other PV-SDHW systems. The savings rate is defined as follows:

1

sav aux conv F E E = - (22)
With,

Eaux, auxiliary heating consumption, including the parasitic energy for the collector loop (circulation pump and the control unit), [kWh];

Econv, energy consumption of a conventional system without the solar component, [kWh].

The highest savings rate (0.59) is obtained with the PV-SDHW system using a solar pump because of the photovoltaic combination and the removal of the controller. From this point of view, the system with the solar pump is more advantageous than the matched flow system with a ∆T of 5 °C.

However, the absence of controller in the PV-SDHW systems can lead to cooling the storage tank in the early morning and late afternoon. The heat losses in the morning (7:00-9:00 am) and late afternoon (5:00-7:00 pm)

were estimated. They are determined as a percentage of the solar energy and correspond to a period when the tank is cooled by the solar heat exchanger. The losses in the morning with the standard DC circulation pump are almost nil because of a relatively late start-up. The solar circulation pump presents the greatest heat loss in the morning (about 1.9%), but this configuration is the most powerful over the year as shown in Fig. 16. The system with LCB presents heat losses in the afternoons that are somewhat higher than with other systems. The heat losses decrease as the temperature difference increases for the matched flow rate systems.

Conclusions

Different PV-assisted SDHW configurations were studied experimentally and numerically based on a developing new circulation pump model. The experimental results showed that the solar circulation pump is more efficient than the standard circulation pump which start-up late in the morning at a solar radiation of 350 W/m 2 . However, the design of solar still needs to optimize because of the excessive output-input temperature difference at the collector. It is also necessary to develop a detailed circulation pump model regarding the correct phase and behavior of the start-up and between morning and afternoon, to precisely assess the energy performance through simulation.

A model of the circulation pump was developed considering the three start-up steps and integrating the model of the electronic devices such as MPPT and LCB. The results of the simulations are in a good agreement with the experimental values. The greatest differences are often occurred early in the morning during start-up.

Figure 1 :

 1 Figure 1: Experimental set-up at INES Figure 2: Solar thermal collectors and PV module Figure 3: Storage tank installed in the test space Concerning the PV-SDHW systems, two DC circulation pumps were studied. The first is a standard DC circulation pump. The second is a commercialized solar circulation pump (Appendix I). They are both composed of a centrifugal pump and a brushless DC motor. The solar pump is designed to connect directly to the photovoltaic module. The integrated MPPT (maximum power point tracker) draws the maximum output from the PV module. In addition, the internal electronic control device and the capacitor start the pump at low solar radiation. Another electronic device, the LCB (linear current booster), was also tested to determine its effect during the start-up phase. The following three PV-SDHW configurations were studied (Fig. 4): -Configuration I: PV + standard DC circulation pump (direct coupling); -Configuration II: PV + solar circulation pump (direct coupling); -Configuration III: PV + LCB + standard DC circulation pump (indirect coupling).

Figure 4 :

 4 Figure 4: Configurations tested

Figure 5 :

 5 Figure 5: Circulation pump voltage and current over time For the configuration incorporating a LCB, Fig. 6 shows the current and voltage of the PV module and circulation pump for one day. It can be seen that the voltage of the PV module remains at 15 V (LCB reference voltage). The voltage and current at the pump vary greatly in the morning until 11:00 am because of cloud cover.

Figure 6 :

 6 Figure 6: Circulation pump electrical behavior Fig. 7 shows the operating points of each configuration obtained by the intersection of the IV (IV represents 'Current' and 'Voltage' characteristics of the PV module with the characteristics of the circulation pump). The characteristics of the PV module are obtained from experimental measurements. The operating points of the solar circulation pump are close to the maximum power points of the PV module (Pmax) through the integration of the MPPT, while those in the standard DC circulation pump are relatively far from the Pmax, between 6 and 10 V. Regarding the configuration using a LCB, the voltage across the PV module remains constant and equls to the LCB reference voltage (14.5 V). The voltage across the pump increases with the solar radiation in this configuration.

Figure 7 :

 7 Figure 7: Operating points of different configurations

Figure 8 :

 8 Figure 8: Comparison of the pump flow rate as a function of the solar radiation

Figure 10 :

 10 Figure 10: Equivalent electrical diagram of the brushless DC motor

A 1 ,

 1 motor static friction coefficient,[Nm] B 1 , motor viscous friction coefficient, [Nm. rad -1 .s] J, motor moment inertia,[kg.m²]

Figure 12 :

 12 Figure 12: Comparison of calculated and measured current of the solar circulation pump Figure 13: Comparison of calculated and measured voltage of the solar circulation pump Fig. 14 shows the changes in the pump flow rate over time. Compared the results from the simulation with

Figure 14 :

 14 Figure 14: Comparison of calculated and measured flow rate in the solar circulation pump In the experimental section, we have shown that, for the same solar radiation, there is, a difference in the pump flow rate between the morning and afternoon because of the change in fluid viscosity. In the circulation pump model, the effect of viscosity in the calculation of the pressure drop coefficient of the solar loop has been considered. An equation incorporating viscosity is used to calculate the pump rotation speed:

Figure 15 :

 15 Figure 15: Comparison of calculated and measured flow rate versus solar radiation

Figure 16 :

 16 Figure 16: Comparison of the energy consumption of different SDHW configurations

Figure 9 :

 9 Figure 1 : Experimental set-up at INES

Figure 11 :

 11 Figure 11 : Circulation pump operation steps

  During thisNally, M., 2007. Viscosity corrections to the pump curve. MC Nally Institute.7-8: 6 p. Seghier, C., 2006. Les émissions de gaz à effet de serre dans les pays industrialisés sont reparties à la hausse.

	Actu-Environnement, 2 p
	Nomenclature
	A 1	motor static friction coefficient	[Nm]
	A 2	Pump torque constant	[Nm]
	B 1	Viscous friction coefficient	[Nm.rad -1 .s]
	B 2	Pump torque constant	[Nm.rad -1 .s]
	C 1	coefficients related to the viscosity, obtained experimentally	[-]
	C 2	coefficients related to the viscosity, obtained experimentally	[-]
	Cp	Specific heat	[J.kg -1 .K -1 ]
	Cυ	Correction coefficient related to viscosity	[-]
	E m	Motor electromotive force	[V]
	E aux	Final energy consumption of the auxiliary heater	[kWh]
	E conv	Final energy consumption of the system without solar component	[kWh]
	G	Solar radiation	[W/m²]
	H	Hydraulic head	[m]
	I a	Motor armature current	[A]
	I PPM	Output current at the maximum power operating point of the PV module (A)	[A]
	I PV	Output current of PV module	[A]
	J	Moment of inertia	[kg.m²]
	K m	Torque constant	[Nm.A -1 ]
	K v	Motor voltage constant	[V.s.rad -1 ]
	L a	Motor inductance	[H]
	l 1	LCB (linear current booster) ratio	[-]
	l 2	LCB ratio	[-]
	Qcol	Flow rate in the solar loop	[l/h]
	R a	Motor armature resistance	[Ω]
	T fluid	Fluid temperature	[K]
	T L	Pump torque load	[Nm]
	T m	Electromagnetic torque	[Nm]
	V sc	PV short-circuit voltage	[V]
	V setpoint	LCB setpoint voltage	[V]
	V m	Motor voltage	[V]
	V PV	Photovoltaic voltage	[V]

Acknowledgments

We wish to thank the APS and CNRS for funding the experimental set-up and the doctoral scholarship.

period, the practical behavior of the circulation pump is not stable because of the control devices integrated in the pump. Regarding the flow profile depending on solar radiation, the model accurately represents the difference between morning and afternoon with the consideration of viscosity change.

Regarding the annual energy performance, the PV-SDHW system with a solar circulation pump has the better performance compared with traditional systems (on-off and matched flow rate systems) and other PV-SDHW configurations. The annual consumption of auxiliary energy (circulation pump and controller) is deleted by coupling photovoltaic with other devices such as the LCB and MPPT, which amounts to approximately 100 kWh. In terms of auxiliary heating consumption in the storage tank, the system using a solar pump reduces 7.5% consumption compared to the system with an on-off controller. Finally, the absence of a controller is not very harmful concerning heat loss at the beginning and end of the day. Also, controlling solar radiation input should improve the performance of in-situ solar installations with the feature of greater simplicity, lower risk of failure, and lower overall energy consumption.

Appendix I

Characteristics of different types of circulation pump

Characteristics


Damiano Brigo

Monique Jeanblanc

Frédéric Vrins


Solmaz Boroomandi Barati

Jean-Chalers Pinoli

Stéphane Valette

Yann Gavet

Nicolas Pionnier

Rémi

Random and geometric model of droplets nucleation on textured surfaces

Introduction

The aim of this study is to simulate dropwise condensation in order to control the rate of droplets nucleation and growth. The process of dropwise condensation attracted lots of attention since about 80 years ago, when Schmidt et al. [1] proved that its heat transfer coefficient is significantly higher than filmwise condensation. Dropwise condensation is preferred when higher rate of heat transfer is needed [2]. The problem that we are dealing with is the formation of liquid droplets in car light shield which will cause the light reflection, in such cases it is preferred to achieve the filmwise regime. In this research we are going to develop a computer algorithm to simulate spatial distribution as well as growth rate of droplets at each time step. The results of the presented algorithm are validated by comparing with the experimental data and spatial distribution in Poisson point process.

Experimental Apparatus

In order to study dropwise condensation, at first a set of gray scale images of droplets nucleation and growth was taken by CCD camera during laboratory experiments. Then these images were binarized and used to drive the information of number and size of droplets at each step. These data is used to validate the results of computer simulation. Experimental setup consists of a chamber containing hot air and cold substrate and a compressor to adjust the relative humidity at about 40%. Temperature of hot air and substrate are set to 86 o f and 62 o f respectively. Nucleation and growth of droplets are recorded at each second by a high resolution CCD camera installed outside the chamber.

Simulation algorithm

The present model is based on physical description of dropwise condensation. At first a specified amount of random points with nucleation site density of 4.6×10 6 ft -2 (the same as experiments) are distributed completely randomly according to Poisson point process. Then each droplet grows through a main loop which consists of three main parts.  Adsorption: which means droplet growth by adsorbing water molecules from humid air. The rate of adsorption can be calculated by the heat balance around a sole droplet according to equation 1.

r k = r k-1 + G, G = 𝑑𝑟 𝑑𝑡 = 𝐶 1 (1- 𝑟 𝑚𝑖𝑛 𝑟 ) 𝐶 2 +𝐶 3 𝑟 , 𝐶 1 = (𝑇 𝑠𝑎𝑡 -𝑇 𝑠𝑢𝑏 ) 𝜌𝐻 𝑓𝑔 , 𝐶 2 = 1 ℎ 𝑖 , 𝐶 1 = 1 𝐾 𝑤 , eq (1)
In this equation r is the droplets radius, k is the counter of droplets generation, G is the growth rate due to adsorption, r min the minimum size of droplets, T sat , air saturation temperature, T sub substrate temperature, ρ water density, H fg water latent heat, h i convection coefficient and K w water thermal conductivity.

 Coalescence: refers to the merging of two or more droplets in order to form a bigger one. This happens when the distance between the droplets is less than the summation of their radiuses.  Nucleation of new droplets: refers to the formation of small droplets that are ready to grow. Nucleation occurs at the beginning as well as during the process.

Results and discussion

Figure 1 shows acceptable accordance between droplets size and nucleation site density for both simulation and experimental data. According to this figure at first due to adsorption droplets grow very rapidly, also coalescence and nucleation of new droplets accelerate the rate of growth which leads to a fast reduction in the number of droplets in earlier stages. After a while due to formation of vacant area between droplets and the friction factor between big droplets and substrate the rate of coalescence reduces. Steady state situation in these graphs comes from the contrary between coalescence and nucleation of new small droplets. Coalescence increases the size of droplets rapidly and reduces nucleation site density, While nucleation of new droplets lower the average droplets size and increase nucleation site densities.

Figure1

  Figure1 Changes in (a)droplet size and (b) Nucleation site density for simulation and experiments

Figure2

  Figure2 Ripley function of progress at steady stateConclusionA physical-based model was used to simulate spatial distribution of droplets in dropwise condensation. The predictions of model for evolution droplets size and nucleation site densities were in good agreement with experimental results. Also the spatial distribution estimated by model was the same as real droplets according to Ripley's function at steady state stages.

Berger et Elise Contraires, for their help in taking experimental results. Supported by the LABEX MANUTECH-