Didier Techer

Damien Banas

Contrasted effects of an anti-cyanobacterial ultrasound device on the non-target freshwater invertebrate species Gammarus roeseli

Keywords: Ultrasound, blooms, water treatment, environmental remediation, Gammarus, biomarkers

The aim of this work was to investigate the effects of an anti-cyanobacterial ultrasound device (supplied by an electrical power of 15 W and emitting at 23 and 46 kHz) on the widespread freshwater amphipod species Gammarus roeseli. First, laboratory scale experiments in 8-L glass tanks showed that an ultrasound exposure of 2 h and 40 min was sufficient to produce 50% mortality, along with a 6.5 °C water temperature increase. Avoiding excessive heating by using a water-cooling and recirculation system permitted an exposure time of 29 h for the same mortality rate. A potential relationship between temperature's rise and amphipod mortality was hence highlighted. Moreover, the use of plastic mesh bag (0.5 mm mesh size) as a physical barrier has not shown any lethal effects of ultrasound exposure. Furthermore, the induction of GPx or GST activity as oxidative stress biomarkers was not observed. This could be explained by reduced ultrasound intensity inside the mesh bags. Thus, according to these results, the tested ultrasound system is not expected to be acutely harmful in the field.

1.

Introduction __________________________________________________________________

In the context of the global warming, cyanobacterial blooms are expected to occur on a more frequent basis in eutrophicated freshwater ecosystems [START_REF] O'neil | The rise of harmful cyanobacteria blooms: the potential roles of eutrophication and climate change[END_REF][START_REF] Paerl | Blooms like it hot[END_REF]. In order to limit excessive microalgal growth and especially harmful algal blooms, a number of so-called "environmentally friendly" physicochemical methods, including ultrasonication, have been developed over the last decades (Jančula and Maršálek 2011; [START_REF] Rajasekhar | A review of the use of sonication to control cyanobacterial blooms[END_REF]. The basic principles explaining the algaecide properties of ultrasonication systems would rely notably on cyanobacteria gas vesicle disruption in conjunction with free radical production following aquatic exposure to ultrasound [START_REF] Rajasekhar | A review of the use of sonication to control cyanobacterial blooms[END_REF]. Such sublethal effects may lead to reduced cell buoyancy and growth and ultimately to cell lysis with potential cyanotoxin release into the surrounding water [START_REF] Rajasekhar | A review of the use of sonication to control cyanobacterial blooms[END_REF]. Hence, in order to evaluate the anti-cyanobacterial efficiency of some commercially available ultrasound transducers, several laboratory and field experiments-often contradictory-have been conducted [START_REF] Wu | The effects of ultrasound on cyanobacteria[END_REF]. As an illustration, Inman (2004) observed a small but significant decrease in total chlorophyll a content during an in situ experiment using a floating low-powered (40 W) ultrasound source operating at a frequency of 28 kHz in a sedimentation lagoon. On the contrary, Lürling and Tolman (2014a) did not find any effect of ultrasound (mix of 20, 28, and 44 kHz with an acoustic power of 0.7 W) on Microcystis aeruginosa (cyanobacteria) and Scenedesmus obliquus (green algae) under laboratory conditions. These authors have rather observed lethal effects on Daphnia magna Straus 1820, a freshwater microcrustacean which was used as a non-target species to assess the environmentally friendly feature of the ultrasound system (Lürling and Tolman 2014a). These results were of great concern since the loss of this zooplankton grazer during the experiments resulted in noticeable algal biomass rise (Lürling and Tolman 2014a). In fact, only few works (mainly focusing on Daphnia) have been conducted so far to assess the potential adverse biological effects of anticyanobacterial ultrasound devices on non-target freshwater organisms (Lürling and Tolman 2014a, b). To our knowledge, no experiments were conducted to In the context of the global warming, cyanobacterial blooms are expected to occur on a more frequent basis in eutrophicated freshwater ecosystems [START_REF] O'neil | The rise of harmful cyanobacteria blooms: the potential roles of eutrophication and climate change[END_REF][START_REF] Paerl | Blooms like it hot[END_REF]. In order to limit excessive microalgal growth and especially harmful algal blooms, a number of so-called "environmentally friendly" physicochemical methods, including ultrasonication, have been developed over the last decades [START_REF] Inman Da | Critical review of actually available chemical compounds for prevention and management of cyanobacterial blooms[END_REF][START_REF] Rajasekhar | A review of the use of sonication to control cyanobacterial blooms[END_REF]. The basic principles explaining the algaecide properties of ultrasonication systems would rely notably on cyanobacteria gas vesicle disruption in conjunction with free radical production following aquatic exposure to ultrasound [START_REF] Rajasekhar | A review of the use of sonication to control cyanobacterial blooms[END_REF]. Such sublethal effects may lead to reduced cell buoyancy and growth and ultimately to cell lysis with potential cyanotoxin release into the surrounding water [START_REF] Rajasekhar | A review of the use of sonication to control cyanobacterial blooms[END_REF]. Hence, in order to evaluate the anti-cyanobacterial efficiency of some commercially available ultrasound transducers, several laboratory and field experiments-often contradictory-have been conducted [START_REF] Wu | The effects of ultrasound on cyanobacteria[END_REF]. As an illustration, Inman (2004) observed a small but significant decrease in total chlorophyll a content during an in situ experiment using a floating low-powered (40 W) ultrasound source operating at a frequency of 28 kHz in a sedimentation lagoon. On the contrary, Lürling and Tolman (2014a) did not find any effect of ultrasound (mix of 20, 28, and 44 kHz with an acoustic power of 0.7 W) on Microcystis aeruginosa (cyanobacteria) and Scenedesmus obliquus (green algae) under laboratory conditions. These authors have rather observed lethal effects on Daphnia magna Straus 1820, a freshwater microcrustacean which was used as a non-target species to assess the environmentally friendly feature of the ultrasound system (Lürling and Tolman 2014a). These results were of great concern since the loss of this zooplankton grazer during the experiments resulted in noticeable algal biomass rise (Lürling and Tolman 2014a). In fact, only few works (mainly focusing on Daphnia) have been conducted so far to assess the potential adverse biological effects of anticyanobacterial ultrasound devices on non-target freshwater organisms (Lürling and Tolman 2014a, b). To our knowledge, no experiments were conducted to assess the ultrasound effect on other invertebrate keystone species of aquatic ecosystem such as Gammarus spp., despite ecological importance of these latter [START_REF] Gerhardt | Gammarus : important taxon in freshwater and marine changing environments[END_REF]) and the growing popularity of water ultrasonic treatment [START_REF] Estévez-Calvar | Potential use of an ultrasound antifouling technology as a ballast water treatment system[END_REF][START_REF] Rajasekhar | A review of the use of sonication to control cyanobacterial blooms[END_REF].

The aim of this work was therefore to investigate the effects of a commercially available ultrasound device (emitting at 23 and 46 kHz) on the widespread amphipod species Gammarus roeseli as a model organism. To achieve this objective, a set of designed experiments were conducted on a laboratory scale. The effect of ultrasound on water temperature and subsequently Gammarus mortality as well as the use of fine mesh bags as physical barriers was thus investigated, revealing for the first time the importance of these latter in limiting ultrasound effects.

2.

Material and Methods __________________________________________________________

Ethical statement

All applied treatments on G. roeseli specimen were in compliance with the general guidelines of the Council of European Communities (European Union No /63/CEE 2010) and the French Animal Care Guidelines.

Ultrasound exposure system

Two transducers were placed adjacent one to another (~ 10 cm between them) emitting continuous signals at an average frequency of 23 and 46 kHz, respectively. They were connected to a function generator, which was used to drive the ultrasound transducers with a power of 15 W (Fig. 1). Details about the transducers' directivity (unidirectional), their emission level and the characterization of the theoretical signal resulting from the combination of [23/46 kHz] were previously described in [START_REF] Techer | Sublethal effect assessment of a low-power and dual-frequency anti-cyanobacterial ultrasound device on the common carp (Cyprinus Carpio): a field study[END_REF]. According to the manufacturer, this ultrasound device has been designed especially for microalgae growth inhibition (SOFCHEM; TWIN-f® Ultrasonic system). The measurement of the resulting emitted acoustic signals in water was carried out using a Brüel and Kjaer Type 8103 hydrophone (omnidirectional, with a frequency range from 0.1.103 to 180 kHz and a receiving sensitivity of  211 dB re 1 V/μPa) connected to a digital multimeter (FLUKE Instruments) [START_REF] Techer | Sublethal effect assessment of a low-power and dual-frequency anti-cyanobacterial ultrasound device on the common carp (Cyprinus Carpio): a field study[END_REF]. A peak-to-peak voltage amplitude of sinusoidal signals output from the ultrasound devices could be measured in water, allowing for comparative evaluation of ultrasound frequency and intensity (recorded by the hydrophone coupled to the multimeter and expressed as voltage) depending on the hydrophone position from the transducers in the glass tanks.

Experimental design

Specimens of Gammarus roeseli were collected and handled according to the procedure described in [START_REF] Sornom | Effects of sublethal cadmium exposure on antipredator behavioural and antitoxic responses in the invasive amphipod Dikerogammarus villosus[END_REF]. Briefly, the day before each set of experiments, specimens of around 15 mm (corresponding to intermediate-sized adults) were collected by handnets (5 mm mesh size) in vegetated areas from a reference site [START_REF] Techer | Allelopathic potential and ecotoxicity evaluation of gallic and nonanoic acids to prevent cyanobacterial growth in lentic systems: a preliminary mesocosm study[END_REF]) in a nonpolluted area (Bwild Moselle River,Tonnoy,54,210,France). They were quickly brought to the laboratory (within 1 h of sampling) in 20 L opaque plastic containers filled with water taken from site. Some leaves and small branches were also added into each tank in order to minimize induced stress by handling and transport. In the laboratory, G. roeseli specimens were maintained in the dark at room temperature in those tanks in continuously aerated water (collected from the reference site) until the beginning of experiments (the next day). The invertebrates were thereafter gently transferred from plastic containers to exposure glass tanks using soft forceps with a blunt end (especially designed for handling small invertebrates). Test organisms were not fed during the exposures.

A set of three experiments involving different conditions was consecutively conducted: (i) uncontrolled physical parameter, (ü) the control of water temperature as a possible bias acting as an environmental stressor (other than ultrasound) on Gammarus mortality, and (üi) the use of an immersed plastic net container to mimic the potential effects of a physical barrier (Fig. 2):

-experiment #1 (Fig. 2a): 120 G. roeseli individuals were randomly and equally distributed into four glass tanks (28 × 14.5 × 20.5 cm), each containing 8 L of clear water (from the reference site). Two tanks were exposed continuously and simultaneously to ultrasound (one transducer of 23 kHz and one of 46 kHz directly immersed into each tank), whereas two others were used as control (no ultrasound). All tanks were packed into polystyrene box to prevent from ultrasound propagation. This experiment lasted 6 h.

-experiment #2 (Fig. 2b): six glass tanks (with 20 Gammarus per tank) including a water-recirculating system between all the tanks (using aquarium pumps) were set up. Water running out of the tanks was collected into a big water supply where it was also regularly cooled (following the immersion of ice cube) before flowing back into the tanks. By this way, water temperature could be maintained not beyond 18.1 °C and was identical into ultrasound exposed as well as control tanks. This experiment ended up after 51h.

-experiment #3 (Fig. 2c): it involved the same recirculating water system, with the exception of two glasstanks per treatment, and all Gammarus (26 per tank) were previously encaged into synthetic bags (all together inside net bags) allowing for freely swimming and constructed from nylon mesh of 0.5 × 0.5 mm mesh size (0.025 mm2). This experiment finished after 7 days.

Each glass tank was inspected at the same time of water physicochemical parameter measurement. Moreover, the frequency of visual observations depended on invertebrate behavior. Dead Gammarus specimens were regularly counted during all exposure period for each experiment. Apparently dead individuals were observed under a binocular microscope, and mortality was defined as failure to respond to gentle mechanical stimulation [START_REF] Taylor | Toxicity of four common pollutants to the freshwater macroinvertebrates Chironomus riparius Meigen (Insecta: Diptera) and Gammarus pulex (L.) (Crustacea: Amphipoda)[END_REF] in the absence of any other signs of physiological activity (ventilation or heartbeat).

Water physicochemical parameters

Physicochemical parameters, i.e., temperature, dissolved oxygen, pH, and conductivity were monitored at regular intervals in each glass tank (in the same location between the different tanks) using appropriate digital probes directly immersed into water (PONSEL ODEON® Digital Handheld Multiparameter).

Sublethal effect assessment-biochemical analyses

Briefly, during the third set of experiments, six specimens per tank (n =12 per treatment, i.e., ultrasound vs control) were randomly collected (after 4 days of ultrasound exposure) and processed for total tissue homogenization [START_REF] Techer | Allelopathic potential and ecotoxicity evaluation of gallic and nonanoic acids to prevent cyanobacterial growth in lentic systems: a preliminary mesocosm study[END_REF] followed by total protein measurement (using the BCATM Protein Assay (PIERCE)), and enzymatic activity determination (total glutathione peroxydase (GPx) and glutathione S-transferase (GST)). GPx and GST activities were selected as potential biomarkers of sublethal effects such as oxidative stress after exposure to ultrasound [START_REF] Techer | Sublethal effect assessment of a low-power and dual-frequency anti-cyanobacterial ultrasound device on the common carp (Cyprinus Carpio): a field study[END_REF]. GPx and GST activities were estimated according to the protocol of [START_REF] Paglia | Studies on the quantitative and qualitative characterization of erythrocyte glutathione peroxidase[END_REF] and [START_REF] Habig | Assays for differentiation of glutathione S-transferases[END_REF], respectively, adapted to microtiter plate [START_REF] Techer | Allelopathic potential and ecotoxicity evaluation of gallic and nonanoic acids to prevent cyanobacterial growth in lentic systems: a preliminary mesocosm study[END_REF].

Statistical treatment

The mean lethal ultrasound exposure duration leading to 50% Gammarus mortality (here referred as LD50) was determined for each ultrasound exposure experiment. For this purpose, continuous counting of dead specimen during exposure was made for the set of the three laboratory tests. Mean LD50 values (expressed in hours ± 95% confidence intervals) were estimated using REGTOX macro Excel (v.7.0.6) based on non-linear regression (Hill model) [START_REF] Vindimian | MS excel macro Regtox.7.0.6 freely available from Eric Vindimian[END_REF]. For enzyme activity determination (experiment #3), all data were expressed as mean ± SE. Normality and homogeneity of variance were tested with the Kolmogorov-Smirnov test and Levene's test, respectively. Significant difference between treatments (ultrasound vs control) was checked by the Student t test. All statistical analyses were carried out with an overall significance level of 0.05 using Statistica v.10.

3.

Results and Discussion ________________________________________________________

During the first experiment (Fig. 1a), there was no waterrecirculating system. Physicochemical parameters were constant in all exposure tanks except temperature. Indeed, in ultrasound tanks, temperature rose about 6.5 °C at the end of experiments (Table 1). A slight decrease in dissolved oxygen could be also reported in ultrasound treated tanks, presumably due to ultrasonication and concomitant lower oxygen solubility following the rise in temperature (Table 1). Nevertheless, water was always oxygen-saturated (Table 1). The LD50 in ultrasound exposed tanks was established at 161 min ([153; 167] min), i.e., ~ 2 h 40 min (Fig. 3a), whereas no mortality was reported in control tanks. The sharp and substantial temperature increase in tanks exposed to ultrasound during the experiment #1 may explain Gammarus mortality in those containers taking into account the species sensitivity to water temperature [START_REF] Vincent | Étude comparée des températures létales supérieures et du préférendum thermique chez deux espèces d'amphipodes épigés du Centre-Ouest[END_REF]. Other hypotheses would rely on a direct lethal effect of ultrasound or on the interaction of ultrasound exposure and elevated temperature. Therefore, in order to avoid any bias related to environmental stressor other than ultrasound on Gammarus mortality, a water cooling and recirculating system was setup to maintain temperature below 20 °C in exposed and control tanks for the rest of the experiments. As reported in the Table 1, water recirculation (and cooling) effectively limited the rise in temperature while leading to similar water temperature values in all experimental units, from 16.75 (experiment #2) to 18.1 °C (experiment #3), which was in accordance with the species thermal preferendum [START_REF] Vincent | Étude comparée des températures létales supérieures et du préférendum thermique chez deux espèces d'amphipodes épigés du Centre-Ouest[END_REF]. As a consequence, the LD50 in ultrasound exposed tanks during the experiment #2 reached 29 h [26 h; 31 h] (Fig. 3b), clearly suggesting that the temperature increase was the main parameter directly influencing Gammarus mortality in the experiment #1 (LD50 of 2 h 40 min). According to this average delay before observing mortality (i.e., more than 1 day in the experiment #2), the tested ultrasound system could be considered as only slightly acutely harmful for the amphipods under these laboratory conditions. In field exposure conditions such as for instance in a fishpond, one may easily expect that the rise in temperature (as reported in the experiment #1) could be neglected, and the toxicity of ultrasound consequently lowered.

In order to test for the potential effects of a physical barrier mimicking in situ exposure conditions, the experiment #3 was conducted where Gammarus specimen was encaged into bags constructed from fine plastic mesh prior to ultrasound exposure (Fig. 2c). These latter resembled litterbags which are usually used in ecological monitoring studies [START_REF] Gessner | A case for using litter breakdown to assess functional stream integrity[END_REF]. Since no mortality has been observed for ultrasound exposed Gammarus during the first 2 days of the experiment #3 (by contrast with the experiment #1 and #2 when dead individuals could be observed following the first hours of exposure), the experiment #3 duration was thus prolonged up to 7 days. Still, daily observations fora continuous 7-day exposure period to ultrasound did not permit to report any mortality (or signs of morbidity) for the ultrasoundexposed amphipods. These results were in contrast with those recently obtained by (Estévez-Calvar et al. 2017) who reported an overall higher effect of ultrasound (20-22 kHz, 200 W) on the mortality of sea water zooplanktonic species when placed into net containers made of a nylon mesh of 80 μm [START_REF] Estévez-Calvar | Potential use of an ultrasound antifouling technology as a ballast water treatment system[END_REF]. Moreover, in order to get insights into potentially more subtle biochemical changes following ultrasound exposure (in the experiment #3), GPx and GST activities were assessed as biomarkers of sublethal effects involving cellular oxidative stress [START_REF] Techer | Allelopathic potential and ecotoxicity evaluation of gallic and nonanoic acids to prevent cyanobacterial growth in lentic systems: a preliminary mesocosm study[END_REF]. Total GPx levels of 5.6 ± 0.5 U mg1protein and 5.8 ± 0.5 U mg1 protein were reported for non-exposed and exposed organisms, respectively (Fig. 4b), whereas GST activities reached 75.6 ± 7.1 and 86.6 ± 9.1 U mg1protein, respectively (also in non-exposed and exposed organisms) (Fig. 4c). Statistical analyses did not permit to sort out any significant difference between ultrasound-exposed organisms and control specimens (Fig. 4b,c). Therefore, these results suggest ultrasound-absorbing or attenuating properties of the mesh bags leading to no apparent biological effect on ultrasound-exposed Gammarus compared to non-exposed organisms.

These assumptions were reinforced by the measurement of the resulting emitted acoustic signals in ultrasound-exposed tanks using a hydrophone. Indeed, in the absence of any synthetic bag, sinusoidal signals corresponding to ultrasound intensity measurement for a frequency of 46 kHz were in the range of + 475/ 515 mV at 8 cm in front of the transducer, whereas a range of + 11/ 15 mV was recorded with the 23 kHz transducer. By contrast, when using a plastic mesh similar to the one used for Gammarus bag confection, the ultrasound intensity for a frequency of 46 kHz was lowered to + 67/ 105 mV at 8 cm in front of the transducer, whereas only + 6/ 11 mV was obtained with the 23 kHz transducer (Fig. 4a). Despite recorded ultrasound frequencies of 23 and 46 kHz, their overall intensity seemed to be reduced from about two to seven times depending on the type of transducer in front of the synthetic nets. According to [START_REF] Raišutis | Application of the ultrasonic characterization methods for highly attenuating plastic materials[END_REF], the waveform of the ultrasound signal transmitted through a plastic object usually is distorted due to different factors, such as the ultrasound frequency dependent attenuation, phase velocity dispersion, and to some extent diffraction phenomenon, geometry of boundaries of the object etc. As an illustration, the use of some plastic materials such as polyvinylidene fluoride (PVDF) may result in important ultrasonic wave attenuation in relation with the material composition and structure [START_REF] Raišutis | Application of the ultrasonic characterization methods for highly attenuating plastic materials[END_REF]. In this work, the composition of the plastic material used for Gammarus mesh bag confection was not provided by the supplier. Nevertheless, one could perhaps suggest that the used material of the plastic containers was a sort of polyvinyl (maybe fluoride-derived) polymeric material. Moreover, apart from the nature of the plastic container, the geometry of boundaries imposed by the grid of 0.5 mm squared of the synthetic mesh in the present work may have played a substantial role in ultrasound wave diffraction and/or attenuation. Indeed, potential waveform diffraction or distortion phenomenon due to the geometry of the mesh holes may have occurred resulting in intensity attenuation of ultrasound in the studied system (of 15 vs 200 W in the work of [START_REF] Estévez-Calvar | Potential use of an ultrasound antifouling technology as a ballast water treatment system[END_REF]. Further investigations concerning the ultrasound propagation and the spatial distribution of the acoustic pressure in presence of different kind of plastic materials may help in understanding the observed attenuation phenomenon.

4.

Conclusion and perspectives ___________________________________________________

The evaluation of the adverse effects of a dual-frequency ultrasound device on Gammarus roeseli through laboratory scale experiments revealed the rise of temperature as a major biological stressor. Consequently, particular attention must be given to the experimental conditions when evaluating the effects of such ultrasound devices on aquatic invertebrate species. Depending on our results, the tested ultrasound device would not be acutely harmful to Gammarus under environmental conditions. Moreover, the use of a fine mesh bag (0.5 × 0.5 mm mesh size) to mimic a physical barrier clearly attenuated the ultrasound intensity and consequently their adverse effects on the amphipods. Hence, in a context of field use, such an ultrasound device would not be expected to be harmful to freshwater macro-invertebrate species like Gammarus . Three experiments were performed: a experiment #1: two tanks continuously and simultaneously exposed to ultrasound vs two tanks non-exposed(control); all amphipods were freely swimming; b three tanks continuously and simultaneously exposed to ultrasound vs three tanks non-exposed (control); all amphipods were freely swimming, and a water cooling and recirculating system was set up; c two tanks continuously and simultaneously exposed to ultrasound vs two tanks non-exposed (control); all amphipods were encaged into 0.5 mm mesh plastic bags

Figure 1 .Figure 2 .

 12 Figure 1. Picture of the ultrasound device used for the three set of experiments: a function generator driving the ultrasound transducers with a power of 15 W. b ultrasound transducers of respectively 46 and 23 kHz

Figure 3 .

 3 Figure 3. a Illustration of the ultrasound frequency and intensity measurement behind the plastic mesh of 0.5 × 0.5 mm (immersed hydrophone) during emission from the 23 kHz transducer; b total GPx activity comparison in control and exposed organisms in the experiment #3; c total GST activity comparison in control and exposed organisms in the experiment #3. Data are expressed as mean ± SE (n = 12). Significant differences were checked by ANOVA (p< 0.05)

Table 1 . Measurement of water physicochemical parameters in glass tanks before experiments (T0) and at the end of experiments (Tf)

 1 US no ultrasound treatment, (+) US combined ultrasound treatment at 23 and 46 KHz frequencies. Data are expressed as mean ± SD

		Experiment #1: no recirculating water, no	Experiment #2: water recirculation and	Experiment #3: water recirculation and cooling-
		cooling			cooling		plastic mesh bags
		() US		(+) US	() US	(+) US	() US	(+) US
	Temperature (°C)	T0	18.85 ± 0.05 18.50 ± 0.10	14.40 ± 0.00	14.40 ± 0.08	17.90 ± 0.00	17.90 ± 0.00
		Tf	18.95 ± 0.25 25.05 ± 0.35	18.03 ± 0.05	18.10 ± 0.00	16.75 ± 0.05	16.80 ± 0.00
	O2 (mg/L)	T0	9.63 ± 0.02	9.43 ± 0.14	10.46 ± 0.01	10.46 ± 0.01	10.12 ± 0.00	10.52 ± 0.38
		Tf	9.20 ± 0.02	8.71 ± 0.09	9.84 ± 0.05	9.82 ± 0.04	10.15 ± 0.00	10.20 ± 0.03
	O2 (%)	T0	104.7 ± 0.3	103.5 ± 1.1	103.5 ± 0.1	103.6 ± 0.1	107.8 ± 0.0	107.5 ± 0.5
		Tf	100.2 ± 0.1	103.7 ± 2.5	104.8 ± 1.0	105.2 ± 0.4	106.1 ± 0.0	106.4 ± 0.3
	pH	T0	8.10 ± 0.00	8.10 ± 0.00	7.40 ± 0.00	7.43 ± 0.12	7.55 ± 0.05	7.55 ± 0.05
		Tf	8.10 ± 0.00	8.10 ± 0.00	7.87 ± 0.05	7.80 ± 0.00	7.65 ± 0.05	7.70 ± 0.10
	Conductivity (μS/cm)	T0	333 ± 5	334 ± 1	192 ± 0	192 ± 0	205 ± 1	204 ± 0
		Tf	335 ± 3	344 ± 3	205 ± 0	205 ± 0	217 ± 1	217 ± 0
	()

Acknowledgements ____________________________________________________________

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. The authors would also like to thank especially Vincent Dufour for his technical assistance. This work was financially supported by Région Lorraine and SO.GE.EAU. The authors also thank P.O. Jost and J. Jost (Sofchem society) for their valuable help regarding the conception, use, and lending of ultrasonic devices.

5.


¸ois Franc

Alain Fillaux
email: fillaux@glvt-cnrs.fr

Matthias J Cousson

Gutmann

Alain Cousson
email: alain-f.cousson@cea.fr

Matthias J Gutmann
email: m.j.gutmann@rl.ac.uk

The Macroscopic Quantum Behavior of Protons in the KHCO 3 Crystal: Theory and Experiments

For hydrogen bonded crystals exhibiting proton transfer along hydrogen bonds, namely O1 -

there is a dichotomy of interpretation consisting in that while the crystal lattice is a quantum object with discrete vibrational states, protons are represented by a statistical distribution of classical particles with definite positions and momenta at any time. We propose an alternative theoretical framework for decoherence-free macroscopic proton states. The translational invariance of the crystal, the adiabatic separation of proton dynamics from that of heavy atoms, the nonlocal nature of proton states, and quantum interferences, are opposed to statistical distributions and semiclassical dynamics. We review neutron scattering studies of the crystal of potassium hydrogen carbonate (KHCO 3 ) supporting the existence of macroscopic quantum correlations, from cryogenic to room temperatures. In addition, quantum fluctuations calculated for superposition states in thermal equilibrium are consistent with measurements of the correlation time. There is no temperature induced transition from the quantum to the classical regime. The crystal can be therefore represented by a state vector and the dichotomy of interpretation must be abandoned.

Introduction

The linear formalism of quantum mechanics extrapolated from the level of electrons and atoms to that of everyday life leads to conclusions totally alien to commonsense, such as Schrödinger's Cat in a superposition of "alive-dead" states and nonlocal observables [START_REF] Einstein | Can quantum-mechanical description of physical reality be considered complete?[END_REF]. Such conflicts lead to a dichotomy of interpretation consisting in that, while at the microscopic level a quantum superposition indicates a lack of definiteness of outcome, at the macroscopic level a similar superposition can be interpreted as simply a measure of the probability of one outcome or the other, one of which is definitely realized for each measurement of the ensemble [START_REF] Bohr | The quantum postulate and recent developments of quantum theory[END_REF][START_REF] Laloë | Quantum mechanics, strange correlations; paradoxes and theorems[END_REF][START_REF] Leggett | Macroscopic quantum systems and the quantum theory of measurement[END_REF][START_REF] Leggett | Testing the limits of quantum mechanics: motivation, state of play, prospects[END_REF][START_REF] Leggett | Dynamics of the dissipative two-state system[END_REF][START_REF] Leggett | Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?[END_REF]. For open systems, this can be legitimated by decoherence [START_REF] Zurek | Decoherence, einselection and the quantum origin of the classical[END_REF] stipulating that an initial superposition state should lose its ability to exhibit quantum interferences via interaction with the environment. However, since the quantum theory does not predict any definite dividing line between quantal and classical regimes, macroscopic quantum behavior is possible for systems decoupled from, or very weakly coupled to, the surroundings [START_REF] Caldeira | Quantum tunnelling in a dissipative system[END_REF]. In principle, there is no upper limit in size, complexity, and temperature, beyond which such systems should be doomed to classicality.

For example, it is a matter of fact that defect-free crystals are macroscopic quantum systems with discrete phonon states at any temperature below melting or decomposition. This is an unavoidable consequence of the translational invariance of the lattice. However, the dichotomy of interpretation arises for the so-called "proton disorder" in crystals containing O-H• • •O hydrogen bonds. The coexistence of two configurations at thermal equilibrium, say O1 -H • • • O2 and O1 • • • H -O2, has been thoroughly investigated in many systems [START_REF] Schuster | The hydrogen bond[END_REF]. Although the light mass of protons suggests that dynamics should be quantum in nature, semiclassical approaches are widely used to rationalize correlation times measured with solid-state NMR and quasi-elastic neutron scattering (QENS). Semiclassical protons are thought of as dimensionless particles, with definite positions and momenta, moving in a doublewells coupled to an incoherent thermal bath. These protons undergo uncorrelated jumps over the barrier and "incoherent tunneling" through the barrier. In fact, these models describe a liquid-like surroundings, at variance with the spatial periodicity of the crystal, and strong interaction with the thermal bath is supposed to lead to fast decoherence. By contrast, vibrational spectra provide unquestionable evidences that the translational invariance and the quantum nature of lattice dynamics are not destroyed by proton transfer. Our purpose is therefore to elaborate a purely quantum rationale avoiding any mixture of quantum and classical regimes.

We shall concentrate on the crystal of potassium hydrogen carbonate (KHCO 3 ) composed of centrosymmetric dimers of hydrogen bonded carbonate ions (HCO - 3 ) 2 separated by K + entities. At elevated temperatures, the coexistence of two configurations for dimers is commonly conceived of as a statistical distribution [START_REF] Benz | Jump motion of deuterons along hydrogen bonds in KDCO 3 . A deuteron relaxation study[END_REF][START_REF] Eckold | Proton disorder and phase transition in KHCO 3[END_REF][START_REF] Thomas | Hydrogen-bond studies. LXXXIV. An X-ray diffraction study of the structures of KHCO 3 and KDCO 3 at 298[END_REF][START_REF] Thomas | Hydrogen bond studies. XCII. Disorder in (HCO 3 ) 2-2 and (DCO 3 ) 2-2 dimers: A neutron diffraction study of KHCO 3 and KDCO 3[END_REF]. By contrast, systematic neutron scattering experiments measuring a large range of the reciprocal space have revealed the macroscopic quantum behavior of protons, from cryogenic to room temperatures, and the theory suggests that this behavior is intrinsic to the crystal state [START_REF] Fillaux | The pauli principle and the vibrational dynamics of protons in solids: A new spinrelated symmetry[END_REF][START_REF] Fillaux | Macroscopic quantum entanglement and "superrigidity" of protons in the KHCO 3 crystal from 30 to 300[END_REF][START_REF] Fillaux | Observation of the dynamical strucuture arising from spatially extended quantum entanglement and long-lived quantum coherence in the KHCO 3 crystal[END_REF][START_REF] Ikeda | Incoherent-elastic-neutron scattering study of the vibrational dynamics and spin-related symmetry of protons in the KHCO 3 crystal[END_REF]. The present contribution is a prelimi-nary attempt to elaborate a consistent presentation of experimental and theoretical works currently in progress.

In Sec. 2 we present the crystal structure and the thermally activated interconversion of dimers. We emphasize why this crystal is unique to observing macroscopic quantum effects. In Sec. 3, we show that the adiabatic separation of proton dynamics leads to decoherence-free states. Then, we introduce the theoretical framework for macroscopic proton states in Sec. 4 and the double-well for protons in Sec. 5. In Sec. 6, the calculated scattering cross-section allows us to interpret neutron scattering experiments in terms of quantum correlations. In Sec. 7, quantum beats arising from the superposition of macroscopic proton states in thermal equilibrium are compared with the correlation time determined with QENS. In the conclusion, we emphasize that the crystal is a macroscopic quantum object that can be represented by a state vector.

2 The crystal structure of KHCO 3 The crystal at 14 K is monoclinic, space group P2 1 /a (C 5 2h ), with four equivalent KHCO 3 entities per unit cell (Fig. 1). Centrosymmetric dimers (HCO - 3 ) 2 linked by moderately strong OH• • •O hydrogen bonds, with lengths R O•••O ≈ 2.58 Å, are well separated by K + ions. All dimers lie practically in (103) planes, hydrogen bonds are virtually parallel to each other, and all protons are crystallographically equivalent (indistinguishable). This crystal is unique to probing proton dynamics along directions x, y, z, parallel to the stretching (νOH), the in-plane bending (δ OH), and the out-of-plane bending (γOH) vibrational modes, respectively. From 14 K to 300 K, there is no structural phase transition. The increase of the unit cell dimensions and of the hydrogen bond length are marginal, but the population of proton sites changes significantly (Fig. 2). Below ≈ 150 K, all dimers are in a unique configuration, say L. At elevated temperatures, protons are progressively transferred along the hydrogen bonds to the less favored sites (configuration R) at ≈ 0.6 Å from the main position. The center of symmetry is preserved and all proton sites remain indistinguishable. There is a general agreement that the population of the less favored site (or interconversion degree ρ) is determined by an asymmetric double-well potential along the hydrogen bonds [START_REF] Benz | Jump motion of deuterons along hydrogen bonds in KDCO 3 . A deuteron relaxation study[END_REF][START_REF] Eckold | Proton disorder and phase transition in KHCO 3[END_REF][START_REF] Fillaux | Calculation of infrared and raman band profiles of strong hydrogen bonds. OH stretching band and proton dynamics in crystalline potassium hydrogen carbonate[END_REF][START_REF] Fillaux | Proton dynamics in the hydrogen bond. the inelastic neutron scattering spectrum of potassium hydrogen carbonate at 5[END_REF], but an in-depth examination of proton dynamics is necessary to distinguishing statistical disorder or quantum delocalization.

The adiabatic separation

Within the framework of the Born-Oppenheimer approximation, the vibrational Hamiltonian can be partitioned as

H v = H H + H at + C Hat , (1)
where H H and H at represent the sublattices of protons (H + ) and heavy atoms, respectively, while C Hat couples the subsystems. For OH• • •O hydrogen bonds, coupling terms between OH and O• • •O degrees of freedom are rather large [START_REF] Novak | Hydrogen bonding in solids. Correlation of spectroscopic and crystallographic data[END_REF][START_REF] Schuster | The hydrogen bond[END_REF], and beyond the framework of the perturbation theory. Two approaches, either semiclassical or quantum, are commonly envisaged.

In the semiclassical view, protons are thought of as dimensionless particles, with definite positions and momenta, moving across a potential hypersurface [START_REF] Benderskii | Tunneling splittings in vibrational spectra of non-rigid molecules: Ix. malonaldehyde and its isotopomers as a test case for fully coupled multidimensional tunneling dynamics[END_REF][START_REF] Benderskii | Tunneling splittings in vibrational spectra of non-rigid molecules. X. Reaction path Hamiltonian as zero-order approximation[END_REF][START_REF] Giese | Multidimensional quantum dynamics and infrared spectroscopy of hydrogen bonds[END_REF][START_REF] Smedarchina | Tunneling dynamics of double proton transfer in formic acid and benzoic acid dimers[END_REF][START_REF] Tautermann | The ground-state tunneling splitting of various carboxylic acid dimers[END_REF]. Complex trajectories involving heavy atoms lead to mass renormalization, and to incoherent phonon-assisted tunnelling [START_REF] Benz | Jump motion of deuterons along hydrogen bonds in KDCO 3 . A deuteron relaxation study[END_REF][START_REF] Eckold | Proton disorder and phase transition in KHCO 3[END_REF][START_REF] Skinner | Proton transfer in benzoic acid crystals: A chemical spinboson problem. Theoretical analysis of nuclear magnetic resonance, neutron scattering, and optical experiments[END_REF]. This approach is quite natural when the Born-Oppenheimer surface is calculated from first principles, but quantum effects can be severely underestimated.

Alternatively, if the classical concept of "trajectory", totally alien to quantum mechanics, is abandoned, adiabatic separation of the two subsystems, namely H H and H at , may lead to tractable models [START_REF] Fillaux | Macroscopic quantum entanglement and "superrigidity" of protons in the KHCO 3 crystal from 30 to 300[END_REF][START_REF] Fillaux | Extended tunnelling states in the benzoic acid crystal: Infrared and raman spectra of the OH and OD stretching modes[END_REF][START_REF] Giese | Multidimensional quantum dynamics and infrared spectroscopy of hydrogen bonds[END_REF][START_REF] Maréchal | Infrared spectra of H-bonded systems[END_REF][START_REF] Schuster | The hydrogen bond[END_REF][START_REF] Witkowski | Infrared spectra of the hydrogen-bonded carboxylic acids[END_REF]. Then, light protons in a definite eigen state should remain in the same state in the course of time, while heavy atoms oscillate slowly, in an adiabatic hyperpotential depending on the proton state, through the coupling term. This separation is relevant for KHCO 3 because adiabatic potentials for different protons states do not intersect each other. Then, protons are bare fermions and quantum correlations should occur [START_REF] Fillaux | Macroscopic quantum entanglement and "superrigidity" of protons in the KHCO 3 crystal from 30 to 300[END_REF].

In fact, the separation is rigorously exact in the ground state, since protons should remain in this state for ever, if there is no external perturbation. Furthermore, for asymmetric double-wells, with wave functions largely localized in each well (see below Sec. 5), the adiabatic separation should also hold for the lowest state of the upper minimum and long-lived superposition states should interfere.

Macroscopic proton states

Consider a crystal composed of very large numbers N a , N b , N c (N = N a N b N c ) of unit cells labelled j, k, l, along crystal axes (a), (b), (c), respectively. The two dimers per unit cell are indexed as j, k, l and j ′ , k, l, respectively, with j = j ′ . For centrosymmetric dimers, there is no permanent dipolar interaction, so that interdimer coupling terms and phonon dispersion are negligible [START_REF] Fillaux | Proton dynamics in the hydrogen bond. the inelastic neutron scattering spectrum of potassium hydrogen carbonate at 5[END_REF][START_REF] Ikeda | Inelastic neutron scattering study of the localized dynamics of protons in KHCO 3 single crystals[END_REF][START_REF] Kashida | Inelastic neutron scattering study of KHCO 3[END_REF]. The eigen states of the sublattice of protons can be therefore represented in a rather simple way with the basis sets of eigen states for isolated dimers A H1-H2 dimer is modelled with coupled centrosymmetric collinear oscillators in three dimensions, along coordinates α 1jkl and α 2jkl (α = x, y, z), with respect to the center at α 0jkl . The mass-conserving normal coordinates independent of j, k, l, and their conjugated momenta,

α s = 1 √ 2 (α 1 -α 2 + 2α 0 ), P sα = 1 √ 2 (P 1α -P 2α ), α a = 1 √ 2 (α 1 + α 2 ), P aα = 1 √ 2 (P 1α + P 2α ), (2)
lead to uncoupled oscillators at frequencies hω sα and hω aα , respectively, each with m = 1 amu. The difference (hω sα -hω aα ) depends on the coupling term (say λ α ). The wave functions, Ψ a njkl (α a ), Ψ s n ′ jkl (α s -√ 2α 0 ), cannot be factored into wave functions for individual particles, so there is no local information available for these entangled oscillators. Consequently, the degenerate ground state of indistinguishable fermions must be antisymmetrized. For this purpose, the wave function is rewritten as a linear combination of those for permuted oscillators as

Θ 0jkl± = 1 √ 2 ∏ α Ψ a 0jkl (α a ) Ψ s 0jkl (α s - √ 2α 0 ) ±Ψ s 0jkl (α s + √ 2α 0 ) , (3)
and the antisymmetrized state vectors as:

|0 jkl+ ⊗ |S = Θ 0jkl+ ⊗ 1 √ 2 [| ↑ 1 ↓ 2 -| ↓ 1 ↑ 2 ] ; |0 jkl-⊗ |T = |Θ 0jkl-⊗ 1 √ 3 | ↑ 1 ↑ 2 + | ↓ 1 ↓ 2 + 1 √ 2 [| ↑ 1 ↓ 2 + | ↓ 1 ↑ 2 ] . (4)
The oscillators are now entangled in position, momentum, and spin. In contrast to magnetic systems [START_REF] Cowley | Quantum entanglement and neutron scattering experiments[END_REF], there is no level splitting, so the symmetry-related entanglement is energy-free. It is also independent of λ α . Furthermore, contrariwise to Keen and Lovesey [START_REF] Keen | Quantum correlation between protons in potassium bicarbonate[END_REF], or Sugimoto et al. [START_REF] Sugimoto | Effects of entanglement on elastic and inelastic scattering functions for neutron scattering from a pair of nuclei in solids[END_REF], we argue, as an experimental fact, that there is no significant exchange integral for protons separated by ≈ 2.2 Å [START_REF] Fillaux | Comment on "quantum correlations between protons in potassium bicarbonate[END_REF].

Neutron diffraction and spectroscopy show that protons in KHCO 3 are neither delocalized nor itinerant particles and there is no sizable energy band structure.

In quantum mechanics, normal coordinates (2) define nonlocal pseudoprotons (m = 1 amu), say P sjkl and P ajkl , with an internal degree of freedom corresponding to symmetric or antisymmetric displacements of two "half-protons", respectively. Each dimer site is a superposition of two such half-protons. Obviously, pseudoprotons are totally alien to the intuitive conception of particles, based on classical mechanics, but they are the actual observables, whereas individual particles are not.

Consider now the sublattice of protons. The spatial periodicity leads to collective dynamics and nonlocal observables in three dimensions. With the vibrational wave function for the unit cell j, k, l, namely Ξ 0jklτ = Θ 0jklτ ±Θ 0j ′ klτ , where τ = "+" or "-", usual phonon waves can be written as

Ξ 0τ (k) = 1 √ N N c ∑ l=1 N b ∑ k=1 N a ∑ j=1 Ξ 0jklτ exp(ik • L), (5)
where k is the wave vector and L = ja + kb + lc, with the unit cell vectors a, b, c. This equation represents collective dynamics of H-H dimers thought of as composed bosons. This would be correct if the crystal was composed of indistinguishable dimer entities (KHCO 3 ) 2 . However, neutron diffraction shows that the crystal structure is composed of KHCO 3 entities related to each other through the appropriate symmetry operations. The probability density of each atom is equally distributed over all equivalent sites and, conversely, the probability density at each site includes contributions from all indistinguishable atoms of the same kind. Consequently, the sublattice of protons must be thought of as a sublattice of nonlocal indistinguishable fermions and antisymmetrization of the plane waves (5) leads to

k • L ≡ 0 modulo 2π. (6)
Consequently, there is no phonon (no elastic distortion) in the ground state and this symmetry-related "super-rigidity" [START_REF] Fillaux | Macroscopic quantum entanglement and "superrigidity" of protons in the KHCO 3 crystal from 30 to 300[END_REF] is totaly independent of proton-proton interaction. Then, the lattice state vectors in three dimensions can be written as:

|Ξ 0+ (k = 0) ⊗ |S ; |Ξ 0-(k = 0) ⊗ |T . (7)
Each macroscopic state of the sublattice represents a nonlocal pseudoparticle with a mass m = 1 amu, namely a pseudoproton, P a or P s , with a definite spin-symmetry and an occupation number of (4N ) -1 per site. There is no local information available for these entangled states and the wave functions Ξ 0τ (k = 0) represent collec- tive oscillations of the super-rigid lattice as a whole with respect to the center of mass of the crystal. Finally, the ground state of the sublattice is a superposition of the pseudoproton states as:

√ N |Ξ 0+ (k = 0) ⊗ |S ; √ N |Ξ 0-(k = 0) ⊗ |T . (8)
This ground state is intrinsically steady against decoherence. Irradiation by plane waves (photons or neutrons) may single out some excited pseudoprotons. Entanglement in position and momentum is preserved, while the spin-symmetry and superrigidity are destroyed. However, the spin-symmetry is reset automatically after decay to the ground state, presumably on the time-scale of proton dynamics. Consequently, disentanglement reaches a steady regime such that the amount of transitory disentangled states is determined by the ratio of the density-of-states for the surrounding atmosphere and external radiations, on the one hand, and for the crystal, on the other. This ratio is so small that disentangled states are too few to be observed, but they allow the super-rigid sublattice to be at thermal equilibrium with the surroundings, despite the lack of internal dynamics. The main source of disentanglement is actually the thermal population of excited proton states. However, even at room temperature, the thermal population of the first excited state (< 1% for γOH ≈ 1000 cm -1 ) is of little impact to effective measurements.

For the sublattice of bosons in the isomorph crystal of KDCO 3 , (3) and ( 6) are not relevant. There is neither spin-symmetry nor super-rigidity. Dynamics are represented with normal coordinates (2) and phonons [START_REF] Benz | Jump motion of deuterons along hydrogen bonds in KDCO 3 . A deuteron relaxation study[END_REF]. Needless to say, the H and D atoms have the same number of 12 degrees of freedom per unit cell, but the symmetrization postulate shrinks the size of the allowed Hilbert space from ∼ 12 N for bosons to ∼ 12N for fermions.

Proton dynamics

Fig. 3 Potential function and wave functions for the OH stretching mode along the hydrogen bond in the KHCO 3 crystal. V (x) = 374x + 0.4389 × 10 6 + 5516 exp(-30.8x 2 ). V and x are in cm -1 and Å units, respectively [START_REF] Fillaux | Calculation of infrared and raman band profiles of strong hydrogen bonds. OH stretching band and proton dynamics in crystalline potassium hydrogen carbonate[END_REF][START_REF] Fillaux | Proton dynamics in the hydrogen bond. the inelastic neutron scattering spectrum of potassium hydrogen carbonate at 5[END_REF]. The oscillator mass is 1 amu.

The interconversion degree at thermal equilibrium (Fig. 2) is determined by the potential function for protons. On the one hand, the bending modes do not play any significant role, since they show rather modest anharmonicity and the population of excited states is negligible. On the other hand, the double-wells for the OH stretching (Fig. 3) is known from experiments. The distance between minima (2x 0 ≈ 0.6 Å) is given by the crystal structure. The upper states at hν 02 and hν 03 were determined from infrared and Raman band profiles [START_REF] Fillaux | Calculation of infrared and raman band profiles of strong hydrogen bonds. OH stretching band and proton dynamics in crystalline potassium hydrogen carbonate[END_REF][START_REF] Fillaux | Theoretical model for calculation of infrared and raman band profiles of strong hydrogen bonds in ordered media[END_REF], and the ground state splitting (hν 01 ) was observed with incoherent inelastic neutron scattering (IINS) [START_REF] Fillaux | Proton dynamics in the hydrogen bond. the inelastic neutron scattering spectrum of potassium hydrogen carbonate at 5[END_REF][START_REF] Kashida | Inelastic neutron scattering study of KHCO 3[END_REF]. The potential obtained through best fitting exercises is over determined and largely model independent. In addition, the oscillator mass of 1 amu is not a free parameter. It is determined by 2x 0 for a given set of energy levels.

For the |0 and |1 states, the potential asymmetry leads to substantial localization of the wave functions in the lower and upper wells, respectively. However, tunneling is possible through the tiny delocalized fraction (ε ≈ 0.05) visible in Fig. 3.

This potential has been a puzzle ever since it was determined because the upper minimum was naively thought of as corresponding to the transfer of a proton. However, this is unlikely, for this would lead to unrealistic dimers composed of di-protonated (H 2 CO 3 ) and non-protonated (CO 2- 3 ) entities [START_REF] Fillaux | Quantum proton transfer and interconversion in the benzoic acid crystal: vibrational spectra, mechanism and theory[END_REF]. Such entities are ruled out by the centrosymmetric character of proton dynamics established by the symmetry-related selection-rules observed in the infrared and Raman [START_REF] Lucazeau | Low temperature Raman spectra of KHCO 3 single crystal[END_REF]. It is now Fig. 4 Schematic view of the tunneling wave functions. For the sake of clarity, the weak component of the wave function in one dimension is multiplied by a factor of 2. clear that, if pseudoprotons are the observables, this nonlocal potential accounts for pseudoproton dynamics along x a or x s . The |0 ←→ |1 transition corresponds to the through-barrier transfer (tunneling) of a pseudoproton as a rigid entity, with no energy transfer to the internal degree of freedom. The IINS bandwidth, very close to the spectrometer resolution [START_REF] Fillaux | Proton dynamics in the hydrogen bond. the inelastic neutron scattering spectrum of potassium hydrogen carbonate at 5[END_REF], shows that this transition is virtually dispersionfree and there is no visible splitting suggesting any difference for the transfer of P a or P s [START_REF] Fillaux | Proton dynamics in the hydrogen bond. the inelastic neutron scattering spectrum of potassium hydrogen carbonate at 5[END_REF][START_REF] Kashida | Inelastic neutron scattering study of KHCO 3[END_REF]. On the other hand, the upper states |2 and |3 correspond to excitations of internal stretching coordinates, ν a (infrared) or ν s (Raman). They are slightly different, but this is unimportant for interconversion since thermal populations are strictly negligible for these states.

The interconversion dynamics involving P a and P s can be rationalized with the potential surface along coordinates x a and x ′ s = x s ± √ 2x 0 :

V (x a , x ′ s ) = V (x a ) +V (x ′ s ). (9)
The energy level scheme comprises three states at 0, hν 01 (with twofold degeneracy) The proton transfer degree calculated supposing the three levels at thermal equilibrium is

ρ(T ) = [p 01 (T ) + 2p 2 01 (T )][1 + p 01 (T ) + p 01 (T ) 2 ] -1 , (10)
where p 01 (T ) = exp(-hν 01 /kT ) is the probability for the transfer of a pseudoproton. The dashed line in Fig. 6 clearly shows that this equation is at variance with observations.

Fig. 6 Temperature effect on the proton transfer degree in the KHCO 3 crystal. Stars: experimental according to [START_REF] Fillaux | Macroscopic quantum entanglement and "superrigidity" of protons in the KHCO 3 crystal from 30 to 300[END_REF]. Solid line: eq. ( 12) for two-levels. Dashed line: eq. ( 10) for three-levels.

In fact, owing to the adiabatic separation of proton dynamics, energy exchange with the surroundings occurs exclusively via photons, with the momentum conservation rule k 10 + k 01 ≡ 0. Consequently, re-entanglement occurs spontaneously in the intermediate state as

|1+, 0-, k 10 + |0+, 1-, k 01 = 2 -1/2 [|0+ |S |0-|T + |1+ |S |1-|T ] (11)
and the transfer degree

ρ(T ) = 2p 2 01 (T )[1 + p 2 01 (T )] -1 , (12)
is in reasonably good agreement with measurements (see the solid line in Fig. 6).

The energy difference of 2hν 01 between RR and LL configurations is therefore confirmed and Figure 6 is an indirect evidence that the degenerate intermediate state observed with IINS is not thermodynamically stable, thanks to the purely quantum re-entanglement mechanism [START_REF] Eckold | Proton disorder and phase transition in KHCO 3[END_REF]. However, the interpretation of Bragg diffraction is ambiguous because the LL and RR configurations of the proton sublattice are crystallographically equivalent, as they are related through a translation vector (a/2, b/2, 0) (see Fig. 5). The reciprocal lattices are identical and it is unknown whether neutrons were diffracted by either sublattice, with probability 1 -ρ and ρ, respectively, or by a superposition state, (1 -ρ) 1/2 |0+ |S |0-|T + ρ 1/2 |1+ |S |1-|T . The former case is a mix- ture of LL and RR configurations analogous to disorder under consideration in many crystallographic [START_REF] Kashida | Structural transition in KHCO 3[END_REF][START_REF] Thomas | Hydrogen-bond studies. LXXXIV. An X-ray diffraction study of the structures of KHCO 3 and KDCO 3 at 298[END_REF][START_REF] Thomas | Hydrogen bond studies. XCII. Disorder in (HCO 3 ) 2-2 and (DCO 3 ) 2-2 dimers: A neutron diffraction study of KHCO 3 and KDCO 3[END_REF], solid-state NMR [START_REF] Benz | Jump motion of deuterons along hydrogen bonds in KDCO 3 . A deuteron relaxation study[END_REF][START_REF] Odin | 13 C and 39 K high-resolution solid-state NMR study of the nonferroic phase transition of potassium hydrogen carbonate. Complementarity between NMR and incoherent neutron scattering[END_REF] and QENS [START_REF] Eckold | Proton disorder and phase transition in KHCO 3[END_REF] works. Alter-natively, a superposition should give rise to quantum interferences corresponding to coherent fluctuations of the probability density at proton sites.

Probing quantum entanglement with neutrons

Neutrons (spin 1/2) are unique to observing the spin-symmetry of macroscopic states [START_REF] Caldeira | Quantum tunnelling in a dissipative system[END_REF]. However, quantum entanglement is extremely fragile, because it is not stabilized by any energy. Consequently, only "noninvasive" experiments, free of measurement-induced decoherence, are appropriate [START_REF] Leggett | Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?[END_REF]. For neutron scattering, this means (i) no energy transfer (ii) no spin-flip and (iii) particular values of the neutron momentum transfer vector Q preserving the super-rigidity. (By definition,

Q = k i -k f ,
where k i and k f are the initial and final wave vectors, respectively.)

The dotted lines in Fig. 1 enhance the network of double-lines of proton sites in dimer planes. We present below neutron scattering experiments for (i) double-lines of protons, (ii) arrays of double-lines in two dimensions, and (iii) the sublattice in three dimensions. For the sake of simplicity, it should be born in mind that elastic scattering events are identical for LL and RR configurations and, therefore, independent of the interconversion degree.

Double-lines of entangled protons

Consider an incoherent elastic neutron scattering (IENS) experiment conducted with (i) the best resolution in energy, in order to reject inelastic scattering events, (ii) a modest resolution in Q, so Bragg peaks merge into a continuum and long-range correlations are overlooked. For momentum transfer Q α along α, the scattering function for an entangled pair (4) can be written as:

S τ i τ f (Q α , ω α ) = ∑ τ f ∑ τ i | 0 jklτ i | exp iQ α α 2jkl -α 0jkl + τ i τ f exp iQ α α 2jkl + α 0jkl |0 jklτ f × 0 jklτ f | exp iQ α α 1jkl -α 0jkl + τ f τ i exp iQ α α 1jkl + α 0jkl |0 jklτ i | 2 × exp(-2W Lα )δ (ω α ). (13)
Each bracket represents a scattering event by a pseudoproton located at both sites (±α 0jkl ). The product of two brackets means that each neutron is scattered simultaneously by the two pseudoprotons superposed at the same sites, either inphase, |± ←→ |± (τ f τ i = +1), or anti-phase, |± ←→ |∓ (τ f τ i = -1). The spin- symmetry is probed along the neutron-spin direction with 100% probability and there is no spin-flip because the initial and final states are |0 jklτ i and |0 jklτ f , respectively, for one scattering event and vice-versa for the other one. Thanks to adiabatic separation, the lattice Debye-Waller factor exp(-2W Lα ) can be factored. The energy transfer is hω α and δ (ω α ) accounts for energy conservation. In the har- monic approximation, the scattering function is [START_REF] Fillaux | The pauli principle and the vibrational dynamics of protons in solids: A new spinrelated symmetry[END_REF][START_REF] Ikeda | Incoherent-elastic-neutron scattering study of the vibrational dynamics and spin-related symmetry of protons in the KHCO 3 crystal[END_REF]

S ±± (Q α , ω α ) = cos 4 (Q α α 0 ) exp - Q 2 α u 2 0α √ 1 + 4λ α + exp -Q 2 α u 2 0α exp(-2W Lα )δ (ω α ), S ±∓ (Q α , ω α ) = sin 4 (Q α α 0 ) exp -Q 2 α u 2 0α 2 √ 1 + 4λ α + u 2 0α 2 exp(-2W Lα )δ (ω α ). (14)
Here, u 2 0α = h/(2mω 0α ) is the mean square amplitude for uncoupled harmonic oscillators in the ground state. The intensity is proportional to the incoherent nuclear cross-section for protons, σ Hi ≈ 80.26 b (1 barn = 10 -24 cm 2 ), and the gaussian-like profiles for uncorrelated scatterers are modulated by cos 4 (Q α α 0 ) and sin 4 (Q α α 0 ) (Fig. 7 a).

Such interference fringes were effectively observed with the MARI spectrometer [1] at the ISIS pulsed-neutron source (Fig. 7 b) [START_REF] Ikeda | Incoherent-elastic-neutron scattering study of the vibrational dynamics and spin-related symmetry of protons in the KHCO 3 crystal[END_REF]. Best fit exercises yield α 0 -values in reasonable accordance with the crystal structure and the estimated os- cillator mass is virtually equal to 1 amu. These experiments are positive evidences of pseudoproton states with spin-symmetry (4). In fact, neutron plane waves are scattered coherently by double-lines of entangled pairs perpendicular to Q α , which are reminiscent of Young's double-slits. However, the interference fringes ( 14) are clearly different from those anticipated for distinguishable (classical) double-slits, which should be proportional to cos 2 (Q α α 0 ) [START_REF] Dürr | Origin of quantum-mechanical complementarity probed by "which-way" experiment in an atom interferometer[END_REF][START_REF] Zeilinger | Single and double-slit diffraction of neutrons[END_REF][START_REF] Zeilinger | Experiment and the foundations of quantum physics[END_REF]. Equation ( 14) is also at variance with the scattering function for protons delocalized in a symmetric double-wells. In this case, there is no definite spin-symmetry for the tunneling states |0+ (ground state) and |0-(at hω t ). The scattering functions for elastic scattering |0± ←→ |0± and inelastic scattering |0± ←→ |0∓ are

S ±± (Q α , ω α ) = cos 2 (Q α α 0 ) exp -Q 2 α u 2 0α -2W Lα δ (ω α ), S ±∓ (Q α , ω α ) = sin 2 (Q α α 0 ) exp -Q 2 α u 2 0α -2W Lα δ (ω α -ω t ). (15)
Then, interferences evidence that a single proton is located in two wells. They are visible if the instrument can effectively resolve the tunnel splitting. Otherwise, complementary fringes would merge into the gaussian profile anticipated for a singlewell.

Clearly, scattering by a superposition of entangled double-lines with spin correlations cannot be confused with other double-slits experiments. The fringes are evidences of nonlocal pseudoprotons and there is no means whatever to probe the local particle behavior.

Since ( 13) and ( 14) hold in the same way for LL and RR configurations, interferences are independent of the interconversion degree. However, at elevated temperatures, the intensity at large Q α -values is depressed by the lattice Debye-Waller factor and fringes are less visible.

Needless to say, interferences are neither expected, nor observed, for KDCO 3 [START_REF] Ikeda | Incoherent-elastic-neutron scattering study of the vibrational dynamics and spin-related symmetry of protons in the KHCO 3 crystal[END_REF].

Diffraction

A necessary condition for noninvasive neutron diffraction is that Q x , Q y , Q z , should match a node of the reciprocal sublattice of protons, so neutrons probe super-rigid states without any induced distortion. The only information conveyed by such events is the perfect periodicity of the sublattice, so the Debye-Waller factor is equal to unity at any temperature. In addition, thanks to the spin-symmetry, the scattered intensity is proportional to the total cross-section σ H ≈ 82.0 b [18, 35]. Otherwise, the spin-symmetry is destroyed, so the intensity scattered by protons is proportional to the coherent cross-section σ Hc ≈ 1.76 b and to the Debye-Waller factor for non- rigid lattices. The enhancement factor σ H /σ Hc ≈ 45 is quite favorable to observing quantum correlations. Furthermore, the intensity scattered by the sublattice of heavy atoms, proportional to σ cKCO 3 ≈ 27.7 b, is depressed by the Debye-Waller factor exp -2W KCO 3 (Q). Therefore, the contribution of heavy atoms at large Q-values is rather weak, compared to that of the entangled sublattice, especially at elevated temperatures.

The dashed lines in Fig. 1 show that proton sites are aligned along x and y, but not along z. Consequently, the noninvasive condition can be realized for Q x and Q y exclusively, whereas Q z never coincides with a nod of the reciprocal lattice of protons. Then, the diffraction pattern depends on whether we consider incoherent or coherent scattering along Q z .

Super-rigid arrays in two dimensions

Consider diffraction by super-rigid arrays in (103) planes and incoherent scattering along Q z . In the unit cell, there are two indistinguishable double-lines parallel to y, so the periodicity of the grating-like structure is D x /2, with D x ≈ a/ cos 42 • ≈ 20.39 Å. On the other hand, the spatial periodicity of double-lines parallel to x is D y = b. The differential cross-section for a superposition of pseudoproton states (7) can be then written as

dσ 2 dΩ ∝ N c ∑ l=1 ∑ τ i ∑ τ f N ′ a ∑ j=1 N b ∑ k=1 exp iQ y kD y -y 0 + τ i τ f exp iQ y kD y + y 0 × [exp iQ x ( jD x /2 -x 0 ) + τ i τ f exp iQ x ( jD x /2 + x 0 )]} 2 2 exp -2W z (Q z ), (16)
with N ′ a = 2N a . Neutrons are scattered either in-phase (τ f τ i = +1) or anti-phase (τ f τ i = -1) by orthogonal pairs of lines separated by 2x 0 ≈ 0.6 Å and 2y 0 ≈ 2.209 Å, respectively. The phase matching condition, namely x 0 (y 0 ) commensurable with D x /2 (D y ), is intrinsic to the crystal structure. {• • •} 2 accounts for simultaneous scattering by the superposed pseudoproton states, without neutron-spin flip. The compound Debye-Waller factor exp -2W z (Q z ), including contributions from all atoms, corresponds to incoherent scattering along Q z . The permutation x 0 ←→ -x 0 , y 0 ←→ -y 0 gives the same equation for RR and LL configurations.

Table 1 Orders n y , n x , and positions Q y , Q x , of rods of intensity arising from the entangled array of orthogonal doubles lines of protons in two dimensions. Obs. Q x : positions in Å-1 of the observed rods of intensity along Q z in Figs 8 and9. Q y D y /π is rounded to integers.

n y Q y ( Å-1 ) Q y D y /π τ i τ f k = Q y /b * Q x Obs. Q x ( Å-1 ) 0 0 0 +1 0 n x π/x 0 0, ±10 1 2.86 5 -1 2.57 (n x + 1/2)π/x 0 ±5, ±15 2
5.71 10 +1 5.14 n x π/x 0 0, ±10 3 8.57 15 -1 7.71

(n x + 1/2)π/x 0 ±5, ±15
The diffraction pattern is composed of rods of diffuse scattering parallel to Q z , cigar-like shaped by the Debye-Waller factor, at Q x , Q y -values corresponding to divergences of [START_REF] Fillaux | Proton transfer in the KHCO 3 and benzoic acid crystals: A quantum view[END_REF]. Such divergences occur at Q y = n y π/y 0 ≈ n y × 2.86 Å-1 (Table 1), since Q y D y /π ≈ 5n y is integer. Contrariwise, there is no divergence for antiphase scattering at Q y = ±(n y + 1/2)π/y 0 , because Q y D y /π ≈ 5(n y + 1/2) is not integer. For n y even, Q y D y /π is also even, τ i = τ f , and ridges are anticipated at

Q x = n x π/x 0 ≈ n x × 10 Å-1 , since Q x D x /π ≈ 68n x
is even. Alternatively, for n y odd, Q y D y /π is also odd, τ i = τ f , and ridges are anticipated at Q x = (n x + 1/2)π/x 0 , since Q x D x /π ≈ 68(n x + 1/2) is even.

The cigar-like shaped rods were effectively observed with the SXD [2, 28] instrument at the ISIS pulsed neutron source (Figs 8 -10). For k = 0, they appear at Q x = 0 and ±(10.00±0.25) Å-1 , in accordance with 2x 0 ≈ 0.6 Å. For k = 1, they are barely visible. For k = 2 or 3, we observe ridges at Q x = ±(5 ± 0.2) and ±(15 ± 0.2) Å-1 , still along Q z . These features are best visible in Fig. 9 for k = 2.6, in accordance with Table 1. There is no visible ridge at k = 4. Then, from k = 5 to 9, we observe in Fig. 8 the same sequence as for k = 0 to 4, and rods at k = 7.7 in Fig. 9. Similar diffuse scattering was observed at low temperature in the k = 0 plane, at 14 K [19] and 30 K [START_REF] Fillaux | Macroscopic quantum entanglement and "superrigidity" of protons in the KHCO 3 crystal from 30 to 300[END_REF] (Fig. 10). As anticipated, the rods are unaffected by the interconversion degree. However, at low temperatures, they are partially hidden by the anisotropic diffuse intensity, centered at Q = 0, due to elastic and inelastic incoherent scattering by protons. This continuum precludes observation of the ridges at ±5 Å-1 for 2 ≤ k ≤ 3. Quite paradoxically, quantum correlations are best visible at elevated temperatures.

As anticipated from Sec. 4, the same experiments performed with a crystal of KDCO 3 do not evidence any cigar-like shaped ridge of enhanced intensity, in addition to regular Bragg peaks, for the sublattice of bosons [START_REF] Fillaux | Observation of the dynamical strucuture arising from spatially extended quantum entanglement and long-lived quantum coherence in the KHCO 3 crystal[END_REF].

Super-rigid arrays in three dimensions

From ( 16), the differential cross-section for the LL or RR pseudoproton states is written as

dσ 3 dΩ ∝ ∑ τ i ∑ τ f N ′ a ∑ j=1 N b ∑ k=1 N c ∑ l=1 exp iQ y kD y -y 0 + τ i τ f exp iQ y kD y + y 0 × [exp iQ x ( jD x /2 -x 0 ) + τ i τ f exp iQ x ( jD x /2 + x 0 )] exp iQ z lD z } 2 2 ( 17
)
This equation describes no spin-flip scattering events that do not destroy the spinsymmetry. Divergences occur along the previous rods of intensity at Q z = ±n z 2π/D z , with D z ≈ c × cos 28 • ≈ 3.28 Å and 2π/D z ≈ 1.92 Å-1 . These enhanced peaks are visible in Figs 8-10, even at rather large Q z -values, thanks to super-rigidity. At elevated temperatures, they clearly emerge from the rods of diffuse scattering [START_REF] Fillaux | Proton transfer in the KHCO 3 and benzoic acid crystals: A quantum view[END_REF] depressed by the Debye-Waller factor. These enhanced peaks were not observed for the deuterated crystal [START_REF] Fillaux | Observation of the dynamical strucuture arising from spatially extended quantum entanglement and long-lived quantum coherence in the KHCO 3 crystal[END_REF].

Experiments presented above, in this present section, are clearly consistent with pseudoprotons forming decoherence-free macroscopic single-particle states with spin-symmetry. The underlying theoretical framework presented in Sec. 4 is based on fundamental laws of quantum mechanics. There is no ad hoc hypothesis or parameter. The adiabatic separation, clearly validated by observations, can be regarded as an intrinsic property of hydrogen bonds in this crystal.

Quantum interferences

Superposition of decoherence-free proton states must lead to quantum interferences, or quantum beats. The non-antisymmetrized wave functions for the states |0 and |1 (Fig. 3) can be written as [START_REF] Fillaux | Quantum proton transfer and interconversion in the benzoic acid crystal: vibrational spectra, mechanism and theory[END_REF]:

Ψ 0jkl = cos φ ψ 0 (x -x m ) + sin φ ψ 0 (x + x m ); Ψ 1jkl = -sin φ ψ 0 (x -x m ) + cos φ ψ 0 (x + x m ); (18)
where x stands for x a or x ′ s and ψ 0 (x ± x m ) are harmonic eigen functions for the second-order expansion of the potential around the minima at ±x m ; tan 2φ = ν 0t /(ν 01 -ν 0t ), where hν 0t ≈ 18 cm -1 is the tunnel splitting for the symmetric po- tential. Then, cos φ ≈ 1 and sin φ = ε ≈ 5 × 10 -2 . Superposition leads to harmonic oscillations of the probability density at the beating frequency ν 0b = 8εν 01 ≈ 4ν 0t ≈ 2.5 × 10 12 , in proton per second units (H s -1 ) [START_REF] Cohen-Tannoudji | Mécanique Quantique[END_REF][START_REF] Fillaux | Proton transfer in the KHCO 3 and benzoic acid crystals: A quantum view[END_REF][START_REF] Fillaux | Quantum proton transfer and interconversion in the benzoic acid crystal: vibrational spectra, mechanism and theory[END_REF].

The LL ←→ RR fluctuation rate can be rationalized with two distinct mechanisms, either single-step or two-stepwise (Fig. 4). The single-step mechanism corresponds to superposition of the states at 0 and 2hν 01 , corresponding to LL and RR configurations (Fig. 5), respectively. The interconversion rate due to quantum beats is: ν 1b = 2εν 0b exp(-2hν 01 /kT ).

For the two-stepwise process, firstly, either P a or P s is transferred at k = 0 to the LR configuration (Fig. 5) with probability exp(-hν 01 /kT ). Secondly, this state undergoes fast re-entanglement [START_REF] Eckold | Proton disorder and phase transition in KHCO 3[END_REF] leading to the upper state (configuration RR) with probability exp(-2hν 01 /kT ). The interconversion rate is then

ν 2b = 2ν 0b exp(-3hν 01 /kT ). (20)
These theoretical rates must be compared to QENS measurements of a KHCO 3 crystal, from 200 to 400 K [START_REF] Eckold | Proton disorder and phase transition in KHCO 3[END_REF]. The scattering geometry (Q x) was selected in order to probe proton dynamics specifically along the hydrogen bonds. The inverse relaxation time (or attempt frequency), τ -1 0 = 2×10 12 s -1 , is sufficiently close to ν 0b to suggest that (i) QENS and vibrational spectroscopy techniques probe the same dynamics and, (ii) the two-stepwise mechanism ( 20) is prevailing, in accordance with the larger pre-factor. In addition, the measured rate follows an Arrhenius law with an activation energy E a = (336 ± 32) cm -1 significantly different from 3hν 01 ≈ 648 cm -1 . In fact, (20) accounts for coherent fluctuations of two pseudoprotons, with probability exp(-3hν 01 /kT ), and pre-factor 2ν 0b , while QENS probes the fluctuation rate of a pseudoproton with probability exp(-3hν 01 /2kT ), and pre-factor ν 0b . Hence, 3hν 01 /2 = 324 cm -1 accords with E a . It transpires that the QENS technique is an incoherent probe of coherent oscillations of the proton probability, because neutrons are plane waves, rather than a particle-like probe of incoherent stochastic jumps [START_REF] Eckold | Proton disorder and phase transition in KHCO 3[END_REF].

In fact, semiclassical models [START_REF] Benz | Jump motion of deuterons along hydrogen bonds in KDCO 3 . A deuteron relaxation study[END_REF][START_REF] Eckold | Proton disorder and phase transition in KHCO 3[END_REF][START_REF] Skinner | Proton transfer in benzoic acid crystals: A chemical spinboson problem. Theoretical analysis of nuclear magnetic resonance, neutron scattering, and optical experiments[END_REF] are based on inappropriate premises. (i) The potential asymmetry supposedly due to static effects of neighbouring dimers should be temperature dependent. This is at variance with the interconversion degree [START_REF] Einstein | Can quantum-mechanical description of physical reality be considered complete?[END_REF] and Fig. 6. (ii) Coupling to phonons is posited to be necessary to mediate the through-barrier proton transfer at low temperatures. This is not relevant within the framework of the adiabatic separation. (iii) A smooth transition to the Arrhenius behaviour of classical jumps is supposed to occur at elevated temperatures. Contrariwise, neutron diffraction shows that there is no transition to the classical regime (Figs 8-10). In addition, [START_REF] Fillaux | Observation of the dynamical strucuture arising from spatially extended quantum entanglement and long-lived quantum coherence in the KHCO 3 crystal[END_REF] and [START_REF] Fillaux | Quantum proton transfer and interconversion in the benzoic acid crystal: vibrational spectra, mechanism and theory[END_REF] show that an Arrhenius behavior is not necessarily an evidence of the semiclassical regime. Logically, these incorrect premises lead to confusing the activation energy E a with the potential barrier [START_REF] Eckold | Proton disorder and phase transition in KHCO 3[END_REF].

Conclusion

It is often argued that a complex system in continuous interaction with its environment should be in a significantly mixed state that cannot be represented by a state vector. In marked contrast to this widespread opinion, we have accumulated consistent experimental evidences that the sublattice of protons can be represented by a state vector at any temperature up to 300 K. This macroscopic object exhibits all features of quantum mechanics: nonlocality, entanglement, superposition and quantum interferences. There is no transition to the classical regime because the plane waves of the thermal bath cannot destroy entanglement intrinsic to the lattice periodicity. The spin-symmetry of proton states can be transitorily destroyed but the decoherence degree is insignificant because the density-of-states of the surroundings is negligible compared to that of the crystal.

The cornerstones of the theoretical framework are: (i) adiabatic separation; (ii) the fermionic nature of protons; (iii) indistinguishability and degeneracy. There is no ad hoc hypothesis or parameter. Entanglement is intrinsic to the crystal symmetry, irrespective of the strength of proton-proton interactions. Dynamics is rationalized with pseudoprotons forming macroscopic single-particle states with remarkable spin-symmetry and super-rigidity. These quantum correlations are effectively probed with neutrons and quantum interferences arising from entangled doublelines or long-range correlations in three dimensions emphasize that protons in the crystal field are not individual particles possessing properties on their own right. Collective dynamics suggest that the whole crystal should be conceived of as a matter field that is a superposition of macroscopic single-pseudoparticle states. In addition, super-rigidity adds a new item, along with superfluidity and superconductivity, to the list of quantum "super" properties in the condensed matter.

The interconversion degree at thermal equilibrium is consistent with the doublewell potentials for pseudoprotons determined from vibrational spectra and quantum beats accord with QENS measurements of the fluctuation rate. The double-wells for protons is invariant over the whole range of timescales from νOH vibrations (∼ 10 -15 s) to diffraction, through QENS, and at any temperature.

This review emphasizes that the dichotomy of semiclassical protons in a quantum crystal lattice should be abandoned. There is every reasons to suppose that this conclusion holds for many hydrogen bonded crystals and macroscopic quantum behaviors open up new vistas for further investigations.

Fig. 1

 1 Fig. 1 Schematic view of the crystalline structure of KHCO 3 at 14 K. Dashed lines through protons are guides for the eyes. a = 15.06(2) Å, b = 5.570(15) Å, c = 3.650(8) Å, β = 103.97(15) • . The ellipsoids represent 50% of the probability density for nuclei.

Fig. 2

 2 Fig. 2 Probability densities for protons in the KHCO 3 crystal at various temperatures, determined with single-crystal neutron diffraction. The ellipsoids represent 50% of the probability density for nuclei.

Fig. 5

 5 Fig. 5 Schematic view of proton configurations for the tunnelling states. In the ground state LL and in the upper state RR at 2hν 01 , protons are fully entangled. In the intermediate state LR (or RL) at hν 01 all proton sites are equally occupied. The spin symmetry and the super-rigidity are destroyed.

Fig. 7

 7 Fig. 7 (a): Comparison of the theoretical profiles S(Q, 0) for a non-entangled pair (dot-dashed), and for an entangled pair of fermions, according to (11) (solid line). The doted and dashed curves represent interference fringes for in-phase and out-of-phase scattering. (b): S(Q y , 0) measured at 20 K for a single-crystal of KHCO 3 (solid curve with error bars). Comparison with the best fit (dashed line) obtained with (14) convoluted with a triangular resolution function. The dashed line with error bars is the difference spectrum. *: Triangular functions due to other scattering events.

Fig. 8

 8 Fig. 8 Cuts of the diffraction pattern of KHCO 3 at 300 K in various (a * , c * ) planes. The arrows emphasize ridges of intensity parallel to Q z and perpendicular to the dimer plane (dash lines along Q x ).

Fig. 9

 9 Fig.[START_REF] Cowley | Quantum entanglement and neutron scattering experiments[END_REF] Diffuse scattering of KHCO 3 at 300 K in between (a * , c * ) reciprocal planes. The arrows emphasize ridges of intensity parallel to Q z and perpendicular to the dimer plane (dashed lines along Q x ).

Fig. 10

 10 Fig. 10 Diffraction pattern of KHCO 3 at 30 K in the (a * , c * ) reciprocal plane at k = 0. The arrows emphasize ridges of intensity parallel to Q z and perpendicular to the dimer plane (dash lines along Q x ).