Romain Gauchon

Jean-Pascal Hermet

Individuals tell a fascinating story: using unsupervised text mining methods to cluster policyholders based on their medical history

Keywords: unsupervised learning, clustering, health insurance, word embedding, Nonnegative Matrix Factorization, Self Organizing Map

Background and objective: Classifying people according to their health profile is crucial in order to propose appropriate treatment. However, the medical diagnosis is sometimes not available. This is for example the case in health insurance, making the proposal of custom prevention plans difficult. When this is the case, an unsupervised clustering method is needed. This article aims to compare three different methods by adapting some text mining methods to the field of health insurance. Also, a new clustering stability measure is proposed in order to compare the stability of the tested processes.

Methods : Nonnegative Matrix Factorization, the word2vec method, and marginalized Stacked Denoising Autoencoders are used and compared in order to create a high-quality input for a clustering method. A self-organizing map is then used to obtain the final clustering. A real health insurance database is used in order to test the methods.

Results: the marginalized Stacked Denoising Autoencoder outperforms the other methods both in stability and result quality with our data.

Conclusions:

The use of text mining methods offers several possibilities to understand the context of any medical act. On a medical database, the process could reveal unexpected correlation between treatment, and thus, pathology. Moreover, this kind of method could

Introduction

Healthcare circuits are a keystone of the medical system in most countries. They represent the succession of treatments every patient goes through. Thus, they tell the story of the patient's medical history : previous and current diseases, the way they have been treated, the effects on the patient's life… Obviously, the current patient's health status depends on past cares and on their efficiency. Thus, studying healthcare circuits could improve the understanding of the reasons for a treatment failure, for example.

It could also indicate how a patient should be treated in the future. For instance, the success of a tertiary prevention action could depend on the different steps the patient has gone through. Thus, the healthcare circuit seems to be a useful input for an algorithm seeking to target a custom prevention plan based on specific patient needs.

Targeting such a prevention plan is a major challenge for health insurers. By improving policyholders' health, they improve their reputation and reduce their health costs efficiently. However, they are not aware of the health status of their policyholders. They are only aware of a crude vision of the healthcare circuit, due to every health expense's being refunded. Thus, health insurers offer a prime example of how healthcare circuit data could be used for medical purposes, since they barely possess other data due to regulations (such as the GDPR).

A way to use this data is to create clusters of people. Classifying is a common practice in the medical field since formulating a diagnosis is already a classification task (e.g. [1], [START_REF] Azar | Fuzzy and hard clustering analysis for thyroid disease[END_REF], [START_REF] Silva | Hybrid analysis for indicating patients with breast cancer using temperature time series[END_REF]). It consists in finding homogeneous groups underlying a given population, in order to later compare two populations or target as a specific cluster for appropriate care.

A clustering method based on the healthcare circuit in the insurance industry has already been proposed in [START_REF] Gauchon | Health-policyholder clustering using health consumption[END_REF]. It is based on a two-steps method : the dimension is first reduced using Non-negative Matrix Factorization (NMF), then individuals are clustered using a selforganizing Kohonen's map. However, this method suffers some limitation: for instance, it cannot create a pregnancy cluster. In order to quantify the stability, a new similarity measure between two clusters is proposed.

Materials and methods

Dimension reduction methods

Dimension reduction makes it possible to identify patterns in a dataset, and thus improves clustering quality by allowing to deal with the dimension curse. These methods have been particularly successful at studying languages: dimension reduction methods are often able to understand general word contexts. It could also improve spectacularly the quality of a clustering method [START_REF] Gauchon | Health-policyholder clustering using health consumption[END_REF]. In this section, we present three dimension reduction methods: NMF, W2V and mSDA. Since these methods are mainly used in the native language processing field, we assume in this section that we work with a text corpus of m different texts counting n different words. The reader could refer to Simpson and Demner-Fushman [6] for a survey of text-mining applications in biomedicine.

Before explaining these methods, we need to introduce a common concept in text mining: frequency matrices. Given a text corpus, a frequency matrix A counts every occurrence of each word for each text. These matrices are usually very large non-negative sparse matrices.

Dimension reduction methods aims to find a matrix A' of dimension m*k, with k<n, minimizing the loss of information from the matrix A.

Non-negative Matrix Factorization

Word2Vec

W2V is a neural network proposed by Mikolov et al. [15]. It aims to create a word vector space of k dimensions, called word embedding, capturing each component of each word and 3 The R package "NMF", developed by Gaujoux and Seoighe has been used to obtain results of this article [START_REF] Gaujoux | A flexible r package for nonnegative matrix factorization[END_REF].

allowing operations on words 4 . This vector space can be seen as a reduction in k dimensions of the initial space. W2V has been used in many fields, such as medical literature study [16],

[17] or text clustering [START_REF] Lilleberg | Support vector machines and word2vec for text classification with semantic features[END_REF].

We implemented the Continuous Bag of Word (CBOW) architecture for this article. It is a neural network with one hidden layer of k neurons. Contrary to mSDA and NMF algorithms, CBOW does not use frequency matrices and focuses on the context of each word: given a studied word, CBOW takes the neighbors of this word as an input.

In order to counter the imbalance between the rare and frequent words, Mikolov et al.

proposed to improve the model by subsampling based on the overall word frequency [START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF].

The neural network is trained to guess the studied word by only knowing its context. Usually, it performs badly at this task. However, the weights of the hidden layer of neurons form the desired word embedding.

Marginalized stacked denoising autoencoders

Autoencoders are usually a kind of neural network trying to compress then reconstruct an input. To do so, the compression must be as efficient as possible. Thus, autoencoders make it possible to capture relationships between data features and to create a reduced dimensional space.

The method called marginalized stacked denoising autoencoders (mSDA) has been proposed by Chen et al. in order to speed up traditional algorithms [START_REF] Chen | Marginalized denoising autoencoders for domain adaptation[END_REF]. Denoising autoencoders work the same way as classical autoencoders, except that the input is deliberately corrupted. For example, it is possible to set each feature to 0 with probability p. Chen et al. pointed out that for this kind of noise, the law of large numbers makes it possible to explicitly determine the result of an infinite amount of corrupted input. Thus, mSDA is a deterministic method and only depends on two parameters: the probability of noise p and the number of stacked autoencoders.

Clustering with self organizing maps

Self organizing map

Self Organizing Maps 5 (SOM) are a clustering neural network architecture popularized by Kohonen [START_REF] Kohonen | The self-organizing map[END_REF]. Its clustering quality is similar to k-means algorithms one, with the added benefit of a natural result visualization. However, results are very sensitive to the input dimension, which justifies the need for a previous dimension reduction step [START_REF] Mingoti | Comparing SOM neural network with Fuzzy c-means, K-means and traditional hierarchical clustering algorithms[END_REF].

In this neural network, each neuron is arranged following a given topology (for example, a two-dimensional hexagonal grid). It is thus possible to define a neighborhood for each neuron, using a so-called neighborhood function h.

Suppose that one wants to cluster M individuals into a m neurons SOM. Each neuron i is initialized with initial random weights . The first individual k is then presented to each neuron, in order to determine the neuron with the weights closest to the individual features, 5 The R package "Kohonen", developed by Wehrens et al. has been used to obtain the results of this article [START_REF] Wehrens | Self-and super-organizing maps in r: the kohonen package[END_REF].

called the Best Machine Unit (BMU). Then the BMU weights, but also neighboring neurons' weights are adjusted, in order to create an area which attracts similar individuals. The process is then iterated.

Once the map is trained, we reduce the cluster number by proceeding to a hierarchical clustering [24].

Projection

The NMF method results in two matrices, one of them being the data projected into a reduced dimensional space. However, W2V only provides an embedding, which can be seen as the basis of the reduced dimensional space. Thus, it is still needed to project the inputs on this space.

In order to do that, we first normalize the basis . We also normalize row by row. We then project all individuals by computing . This is the classical way to project individuals using the Euclidean distance. However, since both the basis and A have been normalized, this is equivalent to projecting individuals using the cosine similarity.

Moreover, mSDA does not directly give an embedding. Instead, it results in a matrix Z with m rows and m+1 columns (the last column being called the bias). To compute the embedding, we have to drop the bias and to take the absolute value of this matrix. This way, we can use a NMF 6 in order to reduce this matrix into a matrix with k rows and m columns, which play the role of H.

From health insurance data to text

A health insurer usually possesses two databases. The first one groups global information about all policyholders, such as age or sex. Due to European data regulation, this database often does not contain a lot of features. For the purpose of focusing only on healthcare circuit, we use this database only to analyze results, and not for clustering.

This database is joined to a second one: the health consumption database. This database contains all the information needed by the health insurer to refund policyholders. It contains the date, the cost and the nature of all policyholder health expenditures (called medical act).

It is then possible to transform these data into a database similar to a text mining dataset.

Each policyholder becomes a text, where words are the reimbursed medical acts, and where text order is given by the chronological order of each medical act. For example, a policyholder using drugs on the 01/02 and hospitalization on the 05/05 becomes the text "drugs hospitalization".

As in text mining, some medical acts are much more common than others (for example, pharmaceutical spending), and a given medical act could carry numerous meanings (e.g.

ultrasound can be used both for maternity and cardiology).

This dataset does not take into account whether two successive medical acts are separated by one day or by one month. To address this point, it is possible to add a "word" capturing the temporal difference. Hence, the previous example can be represented by the text "drugs 3_6_months hospitalization".

As in text mining, it is still possible to construct a frequency matrix, and then to use NMF or mSDA methods. Since some words are so common that they carry almost no information, text mining frequency matrices are usually transformed using tf-idf method 7 . Since we meet the same problems with drugs, and, to a lesser extent, generalist and specialist practitioners, we also implement this pretreatment, which improves results effectively. An example of application of tf-idf for text classification in biomedicine can be found in Srivastava et al [START_REF] Srivastava | Effect of incremental feature enrichment on healthcare text classification system: A machine learning paradigm[END_REF].

Also, as shown by Huang, cosine similarity is more adapted to text comparison than the Euclidean distance [26], thus, henceforth all presented metrics are based on the cosine similarity.

Database : the database used in these studies comes from a private French health insurer. It is composed of 28,540 women aged between 16 and 62 years. Most of them are working women. There are more than 1,300,000 medical acts observed over one year, distributed 7 For the NMF algorithm, taking the logarithm of the frequency matrix can lead to better results [Gau 2019]. Thus, we tested both logarithm and tf-idf preprocessing. We kept the logarithm for the NMF method and tf-idf for W2V and mSDA methods.

into a hundred different medical acts. Women without at least one medical consumption have been removed.

Results

Dimension reduction

The first step of the proposed method consists in creating a reduced dimensional space by obtaining a health consumption embedding. In order to make comparison easier, the chosen dimension of the final space is 20, regardless of the dimension reduction method. By normalizing the embedding by columns, it is possible to see each embedding vector as a Medical Act Clusters (MAC) (see Figure 2). First, the embedding obtained with the W2V dimension reduction is not meaningful. It is due to the faculty of W2V to capture nonlinear relationships between medical acts.

On the contrary, MACs obtained with NMF and mSDA are easily understandable. However, MACs obtained with NMF seem to be less accurate than the ones obtained with mSDA, especially for uncommon medical acts, like "Other optic" or "optical surgery" (consumed less than 200 times).

Both algorithms do not focus on the same effects. For example, NMF clusters hospitalization and physical therapy together, whereas mSDA combines psychiatric and classical hospitalization. As for maternity MACs, NMF is not able to dissociate maternity medical acts from biological analysis or ultrasound. Instead, mSDA is able to dissociate maternity from both ultrasounds and biological analysis. However, it is interesting to notice that mSDA puts psychiatry into maternity MAC, this likely being due to postpartum depression.

Finally, notice that for both algorithms, a same medical act can belong to multiple MACs.

Clustering

The dimension reduction makes it possible to obtain a small number of high quality features before clustering. The SOM method has been chosen to take advantage of its the natural visualization (an example is given in Appendix A). To make the comparison between each algorithm easier, 20 clusters have been done for each classification. For the sake of concision, we call the whole process by the name of "dimension reduction method".

To understand the meaning of a given cluster, it is possible to compute the rate of people consuming these acts for every medical act (Example given in Figure 4). It is of interest to note that W2V gives understandable final results, even if the embedding is not meaningful. However, the clusters created this way usually seem to be less pure. NMF systematically fails to create a maternity cluster, whereas mSDA rarely creates a dentures cluster. All three methods create several day-to-day care clusters. They also create a huge low-consuming policyholder cluster of around 8 000 people, containing healthy women.

There are two main qualities for a clustering method: clustering quality and stability.

Clustering quality are compared using the cluster inertia. Classical similarity measure, such as Gower's distance, cannot be used to compare two partitions. To compare stability, we introduce a new similarity, inspired by the Jaccard coefficient as suggested by Hennig [27].

To the best of the authors knowledge, this similarity has not been seen in this form before. It can be used for a totally unsupervised context, with a nondeterministic algorithm such as a Kohonen's map.

Let N be a set of features. Let and two partitions in k subsets (clusters) of N. Intuitively, for each couple of features in a subset in , the clustering method is stable if they also belong to the same subset in .

Formally : Let (resp ) be the cardinal of subset number i in the partition (resp ).

The number of possible couple in a subset is given by Let be the number of features in the subset number i of and in the subset j of .

We compute . It is easy to verify that sim is a partition similarity.

This way, we can test if two clusters are similar, and thus test if the method is stable. (see Figure 5 for an example). Thus, = 3. Moreover, and

Finally, the similarity beetween the two clusters is .

Since the NMF and mSDA dimension reduction are almost deterministic, we have constructed 10 different SOMs. We have also constructed 5 different embeddings using W2V, and each of them were used to construct 5 different SOMs.

Numerical analysis

Analyzing each cluster is necessary for a better understanding of their underlying population.

Here, we take the example of clusters mainly described by a high consumption of biological tests. Purity score shows the number of people in the clusters consuming biological tests.

mSDA produces a single biological test cluster whereas W2V and NMF produce two biological test clusters.

The mSDA cluster is quite centered on biological test consumption. These profiles could be linked to either chemotherapy control, chronic fatigue or urinary infection. Maternity profiles are contained in another cluster. This is not the case for NMF, which fails to create a maternity cluster. Thus, maternity profiles can be found in the NMF (B) cluster. NMF (A) and W2V (A) are similar to the mSDA cluster, while penalizing optic and denture consumption less. Finally, W2V (B) is probably a cluster of people subject to disease monitoring.

Conclusion :

Three clustering processes based on the use of the healthcare circuit have been presented and compared. All three processes are in twofold: first the dimension is reduced, and then a clustering is performed using a self-organizing map. The dimension reduction step makes it possible for the self-learning process to understand the correlation between medical acts, and creates high quality features for the clustering step. The method based on the W2V principle offer the best results. However, results offered by the one based on mSDA are closed, but this method is much more meaningful. All methods offer a useful clustering and a more precise understanding of the makeup of a population's health.

Moreover, a new stability measure has been proposed, in order to compare all three methods. It results that the mSDA method is the most stable methods for our data.

The method has been tested on a health insurance database. Moreover, no other medical information has been added in the clustering process. However, it would be of interest to use them in a more general medical context, such as for studying and clustering low back pain diseases. Adding features to the clustering algorithm could improve results easily and effectively.

Conflict of interest statement : Jean-Pascal Hermet and Romain Gauchon work for ADDACTIS France, an actuarial consulting firm.

Figure 1 :

 1 Figure 1 : The main stages of the process

Figure 2 :

 2 Figure 2 : simplified display of tested methods.

Figure 3 :

 3 Figure 3 : Dimension reduction leads to Medical Act Clusters. The last line is a vector of the

Figure 4 :

 4 Figure 4 : Comparison of final clusters obtained. A dark square means that most of the

Figure 5 :

 5 Figure 5 : Two clusterings in 3 classes are made on a set of 9 features (A,B,…,I). It appears

used both in text mining [8], [9] and in medicine [10], [11]. Since it has performed better than other algorithms in the field of insurance [4], the snmf/l algorithm 3 as proposed by Kim and Park has been used [12].

	Lee and Seung [7] proposed a method for factorizing the matrix A into a produce of two
	matrices W and H with m (resp. k) rows and k (resp. n) columns. Since it is not always
	possible to find W and H such as	, NMF usually amounts to finding W and H
	minimizing d(WH,A), with d() a function measuring differences between two matrices.
	The matrix H is the basis of the reduced dimensional space. It can be seen as a fuzzy word
	clustering: it represents the frequency of apparition of word pairs in the same text. Since this
	basis is not orthogonal, it makes it possible for a word to belong to two different clusters,
	and thus takes into account the different meanings of a word. Matrix W is the projection of
	matrix A in the reduced dimensional space. W shows if a text is well represented by each
	word cluster created this way.
	NMF has been

Table 1 :

 1 Two stability measures have been computed for W2V : one comparing all 25 SOMs, and one comparing two SOMs only if they are obtained from the same embedding, in order to test the variability caused by the SOM algorithm alone. clustering quality and stability

		NMF mSDA W2V W2V (for a given embedding)
	Stability mean	47% 56% 46%	58%
	R² mean	35% 43% 45%
	R² maximum	36% 45% 48%
	Both mSDA and W2V offer better cluster quality than NMF (in terms of inertia). Moreover,
	mSDA is the most stable process. It is interesting to notice that a big part of W2V's variability
	comes from the dimension reduction, which has also been observed by Baldassini and
	Serrano [5].

Table 2 :

 2 "biological tests" clusters, numerical details. Both NMF and W2V produce two "biological test" clusters.

As an example, Mikolov et al. wrote that in that kind of space: King -Man + Women = Queen.

PCA and SVD have also been tested, resulting in worst results. Also, it is possible to duplicate the rows of the matrix Z a few times with a very small noise, which makes it possible for NMF to find a better correlation between medical acts and result in a better embedding.

Acknowledgements :

The authors would like to thank ADDACTIS France for providing the database. This paper was realized within the framework of the Chair Prevent'Horizon, supported by the risk foundation Louis Bachelier and in partnership with Claude Bernard Lyon 1 University, ADDACTIS France, AG2R La Mondiale, G2S, Covea, Groupama Gan Vie, Groupe Pasteur Mutualité, Harmonie Mutuelle, Humanis Prévoyance and La Mutuelle Générale.


Manuel Pubellier

Mario Aurelio

Benjamin Sautter

The life of a marginal basin depicted in a structural map of the South China Sea

The South China Sea (SCS) is presented here as a case example to demonstrate the evolution of basins developed at convergent boundaries. The structural map published in 2017 by CGMW at the 1:3 million scale allows to visualize the location of the rifting faults from a normal to hyper-extended crust, the shape and structure of the oceanic crust and their late involvement in a convergent margin. It highlights the reactivation of the Mesozoic tectono-stratigraphic setting such as broad folds and granitic plutons during the rifting, and the effect of the resulting architecture on the NW Borneo accretionary wedge.

Introduction

Like most of the basins of SE Asia, the South China Sea (SCS hereafter) developed by rifting from the Palaeogene [START_REF] Taylor | The tectonic evolution of the South China Basin[END_REF] and suffered shortening starting from the Middle Miocene, as a result of the variations of stress imposed by the neighboring subductions. Because of its narrow V-shaped oceanic crust, the wide extension of its rifted continental crust, and the various styles of rifting, the SCS basin is a text-book subject to study the formation of passive margins developed over former subduction zones. In addition, the SCS has been the focus of scientific interest in past decades [START_REF] Katili | Geology of Southeast Asia with particular reference to the South China Sea[END_REF] including ODP (Leg 184) and recent IODP (Leg 349, [START_REF] Li | the Expedition 349 Scientists[END_REF] drillings, oil and gas exploration, and projects from several international teams onshore and offshore, which supplied a great deal of seismic and other geophysical and geological data [START_REF] Franke | The final rifting evolution in the South China Sea[END_REF][START_REF] Sibuet | Geodynamics of the South China Sea[END_REF]. It benefits nowadays from one of the most extensive geophysical coverage in the world. The basin structures have been recently compiled on a structural map by a large group of experts of the region. This map highlights the basement structures related to the Late Cretaceous to Middle Miocene extension as well as the late evolution of the basin, associated with shortening features on its southern side.

The Structural Map

If geological maps depict as accurately as possible the structures and geological formations according to their age, the principle of superposition of strata makes it difficult to see the tectonic events covered by basins, platforms or recent sedimentary deposits. On the other hand, tectonic maps which group into time brackets the corresponding tectonic events or orogenies, also take into account the lithologies and metamorphic conditions. However they become complex when several events are superimposed.

On the Structural Map of the South China Sea [START_REF] Pubellier | Structural map of the South China Sea[END_REF]) based on a large amount of industrial and academic data, an effort was made to select the main structures and lithologies having played a key role in the formation and the evolution of the SCS. The color code was designed in order to facilitate the offshore/onshore correlation of the structural units. The basement rocks older than the Jurassic were not differentiated and an emphasis was given to the Mesozoic granitoids and older magmatic and metamorphic rocks as well as the Cenozoic volcanic edifices onshore and offshore which have been deposited both on the margins and the sea-floor. The main crustal structural entities were distinguished as thick continental basement, stretched continental crust, inferred zones of exhumed mantle (even below the post-rift sediments), volcanic arc crust in the Philippines, and oceanic crust. Sedimentary bodies onshore and offshore are wedges, platforms and basins, as well as the extent of the Mesozoic marine environment. The age of the crust in the basins other than the SCS (e.g., Celebes Sea, Sulu Sea) is not specified. Rifted Cenozoic basins are underlined and post-Mid Miocene basins appear transparent over rifted structures. Instead of showing the picks or the models of the debated magnetic anomalies, the map depicts the non-controversial magnetic anomalies and emphasizes on the structural fabrics of the sea-floor. The faults represented as black normal faults on the map are those associated with the crustal extension since the beginning of the Late Cretaceous. Other faults, which may appear crossing or paralleling the rifting faults are reversed or thrust faults which mostly took place during the Middle Miocene and the Pliocene. The thin red faults emphasize the shallow-rooted faults which character-ized the gravity tectonic processes that affected the thick Miocene deltas of the SCS.

Geological Features Linked to Extensional History

The basement, which is similar in China, Vietnam and part of the western Philippines is undifferentiated on the map. During Mesozoic times, the area was sitting on the upper plate of a subduction zone, resulting in an impressive coverage of Cretaceous granites sometimes separated by narrow Cretaceous molasse basins. These granitic bodies, widespread offshore in the extended crust conditioned the location of the extension via large detachments and normal faults; later cut by steeper faults. The geometries of the faults vary from E-W to NE-SW indicating that the rifting underwent several stages with different stretching directions. The extension started during Late Cretaceous before the rifting sensu stricto which is clearly documented since the Early Eocene only. Stretching and thinning were important and resulted in a wide "Basin and Range" like province [START_REF] Franke | The final rifting evolution in the South China Sea[END_REF][START_REF] Pichot | Deep crustal structure of the conjugate margins of the SW South China Sea from wide-angle refraction seismic data[END_REF] which was sustained near sea level during the entire duration of the rifting process. This province is seen on both margins of the SCS and ultimately exhumed the mantle like in Vietnam and SE of Taiwan. Some faults are low angle detachments and surround the granitic and metamorphic basement. The structure of the margin by the end of the rifting is dominated by a severe boudinage of a thin crust [START_REF] Savva | Different expressions of rifting on the South China Sea margins[END_REF], on which the low-angle normal faults are either the granitoids boundaries or the short limbs of Cretaceous folds. Reactivation of early structures is common during rifting or basin inversion [START_REF] Festa | A Jurassic oceanic core complex in the high-pressure Monviso ophiolite (W Alps, NW Italy)[END_REF][START_REF] Balestro | Role of Late Jurassic intra-oceanic structural inheritance in the Alpine tectonic evolution of the Monviso meta-ophiolite Complex (Western Alps)[END_REF]. Because the thinned margin was sustained at shallow depth, platform and reef carbonates occupy some of the bathymetric highs. Their development occurred mainly during Late Oligocene to Mid Miocene and during Late Miocene times. The oceanic crust of the SCS was represented with the generally accepted main magnetic anomalies [START_REF] Briais | Updated interpretation of magnetic anomalies and seafloor spreading stages in the South China Sea: implications for the Tertiary tectonics of Southeast Asia[END_REF][START_REF] Sibuet | Geodynamics of the South China Sea[END_REF]. The areas where the crust is younger with controversial age is shown with a different color according to recent models. The geometry of the ocean floor basin has a "V" shape configuration, which ends to the SW as a propagator, implying that the age of the crust (and therefore the breakup) may be diachronic. It is likely that part of the NW Sulu Sea is actually floored by an oceanic crust which could represent a relic of the Proto South China Sea, to be correlated with the Palawan ophiolite.

The SCS basin also raises questions about the time of the breakup and the time of cessation of the extension. The spreading of oceanic crust started by 33 Ma in the northern and central part of the basin although rifting continued until at least 15.5 Ma, at a time when spreading was already finishing [START_REF] Franke | The final rifting evolution in the South China Sea[END_REF]. Furthermore extension is also observed in the midst of the oceanic crust as indicated by low angle normal faults. It is only during early Late Miocene times (circa 12 Ma) that extension ceased and regional subsidence was triggered, and marked by a well-known Mid Miocene unconformity (MMU). The unconformable Late Miocene series are indicated on the map; they also seal the late structures of the collision along the NW Borneo Wedge.

Geological Features Linked to Shortening History

When the SCS basin subsided during the Late Miocene, the NW Borneo wedge started to rise quickly resulting in its subsequent erosion. Likewise other places in the world, the reactivation of crustal heterogeneities during extension and their implication in the later shortening [START_REF] Festa | A Jurassic oceanic core complex in the high-pressure Monviso ophiolite (W Alps, NW Italy)[END_REF] is well expressed in the SCS. It has been documented that the compression which marks the end of the subduction of the Proto South China Sea is responsible for the thrusting of the NW Borneo wedge onto the southern margin of the SCS by the Middle Miocene. This wedge is dominated by two large tectonic units which represent the distal sedimentary series (Rajang wedge) and the proximal series (Crocker Wedge) deposited on the shelf of the conjugate block of the SCS (Palawan-Luconia Block). There is evidence that wedging in Middle Miocene (Pagasa Wedge) may have extended to the north in SW Palawan, albeit at a lower magnitude [START_REF] Aurelio | Middle to Late Cenozoic tectonic events in south and central Palawan (Philippines) and their implications to the evolution of the south-eastern margin of South China Sea: evidence from onshore structural and offshore seismic data[END_REF]. The consequence of the probable slab detachment of the subducted Proto South China Sea slab, and the thrusting of the NW Borneo wedge on this later block induced thickening and uplift, which in turn generated sub-aerial conditions for the NW Borneo wedge. The resulting erosion created the large deltas of Champion, Balingan, and Baram, starting from the end of the Early Miocene. Excess of sediment loading in the deltas induced gravity tectonics [START_REF] Sapin | Counterregional normal faults in shale-dominated delta: origin, mechanism and evolution[END_REF]. The resulting gravity provinces are characterized by red faults which include growth faults and toe-thrusts affecting generally the mid-Miocene to Recent sedimentary pile.

Lessons Learned from the SCS Case Study

Detailed geological mapping and the here presented synthesis of geodynamic of the SCS indicates that some accepted concepts of rifted basins must be handled with care. Among these are 1) the varying age of the breakup, highlight that extension may start before the rifting (orogenic collapse of former topography), and may continue long after the rifting has ceased, 2) the long lasting continental or shallow marine conditions without regional subsidence, 3) and the late crustal rebound associated with the end of the neighboring subduction, which may trigger high vertical motion and develop large deltas prone to gravity tectonics.

Manuel

  Pubellier obtained his Ph.D. from the University Pierre & Marie Curie of Paris in 1987. He worked as a lecturer in the Republic of Haiti (Caribbean) in his early days, and joined the French National Center for Scientific Research (CNRS) in 1988 and became a Research Director at the Laboratory of Geology of the Ecole Normale Supérieure de Paris in 2010. He has a long teaching experience as Visiting Professor at the IFP School and in Universities of Paris, Hong Kong, and UTP in Malaysia. His work is in the field of geodynamics applied to basins formation, and the comparison between geophysical and geological observation of the deformation, particularly in SE Asia. He is currently Secretary General of the Commission for the Geological Map of the World (CGMW). Mario Juan A. Aurelio obtained a B.Sc. Geology degree from the University of the Philippines in 1985. He went on to obtain a Ph.D. in Structural Geology and Tectonics in 1992 at the Université Pierre et Marie Curie in Paris, France, followed by a Post Doctorate fellowship on Geodynamics through Space Geodesy and Seismotectonics in France and Germany (1994-1995). Prior to joining the academy, Dr. Aurelio worked for 20 years as a geologist at the Mines and Geosciences Bureau (MGB). He is currently Professor at and the Director of the National Institute of Geological Sciences, University of the Philippines (UP-NIGS). He is an Adviser of the Philippine Delegation to the Annual Assembly of the International Seabed Authority and a member of the Philippine Extended Continental Shelf Technical Working Group. Benjamin Sautter received his Bachelor's degree in Earth Sciences, Environment and Planets from the Institut de Physique du Globe de Paris (IPGP) in 2010, and a Master's degree in Geology and Natural Hazards in 2012. He completed in 2016 a Ph.D. at the Ecole Normale Supérieure de Paris (ENS) in collaboration with the Universiti Teknologi PETRONAS focused on the Late Paleozoic to Present tectonic evolution of the northwestern Malaysia Peninsula. He had previously conducted a study of the geochronology and metamorphism of the internal zone of the Longmen Shan and the cosmogenic dating of alluvial terraces of the Ürümqi River (Tien Shan). His main research interests are Tectonics, Structural Geology and Geochronology. He is now lecturer at the Universiti Teknologi Petronas.

Acknowledgements

With contributions from :Dimitri Savva (ENS), François Sapin (TOTAL), Dieter Franke (BGR), Van Phach Phung (VAST), Jean-Luc Auxiètre, Lung Sang Chan (HKU), Florian Meresse (ENS), Benjamin Sautter (ENS), Virginie Le Gal (ENS), Kristine Taguibao (NIGS), Char-Shine Liu (NTU), Stephan Steuer (BGR), Yazid Mansor (UTP), David Menier (UBS), Kirk Mcintosh (U Texas), Jean Claude Sibuet, Nicolas, Chamot-Rooke (ENS), Van Tri Tran (IGMR-Vietnam), Mazlan Madon (PETRONAS), Herman Darman (SHELL).

Printed version of the map available at: https://ccgm.org/en/36-oceans Price: 20 Euros.

< Remove oceanic crust


Yvan Castin

Alice Sinatra

Hadrien Kurkjian

Landau phonon-roton theory revisited for superfluid helium 4 and Fermi gases

Keywords: numbers: 03.75.Kk, 67.85.Lm, 47.37.+q

Liquid helium and spin-1/2 cold-atom Fermi gases both exhibit in their superfluid phase two distinct types of excitations, gapless phonons and gapped rotons or fermionic pair-breaking excitations. In the long wavelength limit, revising and extending Landau and Khalatnikov's theory initially developed for helium [ZhETF 19, 637 (1949)], we obtain universal expressions for three-and four-body couplings among these two types of excitations. We calculate the corresponding phonon damping rates at low temperature and compare them to those of a pure phononic origin in high-pressure liquid helium and in strongly interacting Fermi gases, paving the way to experimental observations.

Introduction -Homogeneous superfluids with shortrange interactions exhibit, at sufficiently low temperature, phononic excitations φ as the only microscopic degrees of freedom. In this universal limit, all superfluids of this type reduce to a weakly interacting phonon gas with a quasilinear dispersion relation, irrespective of the statistics of the underlying particles and of their interaction strength. Phonon damping then only depends on the dispersion relation close to zero wavenumber (namely, its slope and third derivative) and on the phonon nonlinear coupling, deduced solely from the system equation of state through Landau-Khalatnikov quantum hydrodynamics [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF].

In experiments, however, temperatures are not always low enough to make the dynamics purely phononic. Other elementary excitations can enrich the problem, such as spinless bosonic rotons in liquid helium 4 and spinful fermionic BCS-type pair-breaking excitations in spin-1/2 cold-atom Fermi gases. These excitations, denoted here as γ-quasiparticles, exhibit in both cases an energy gap ∆ > 0. Remarkably, as shown by Landau and Khalatnikov [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF], the phonon-roton coupling, and more generally phonon coupling to all gapped excitations as we shall see, depend to leading order in temperature only on a few parameters of the dispersion relation of the γ-quasiparticles, namely the value of the minimum ∆ and its location k 0 in wavenumber space, their derivatives with respect to density, and the effective mass m * close to k = k 0 . We have discovered however that the φ -γ coupling of Ref. [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF] is not exact, a fact apparently unnoticed in the literature. Our goal here is to complete the result of Ref. [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF], and to quantitatively obtain phonon damping rates due to the φ -γ coupling as functions of temperature, a nontrivial task in the considered strongly interacting systems. We restrict to the collisionless regime ω q τ γ ≫ 1 and ω q τ φ ≫ 1, where ω q is the angular eigenfrequency of the considered phonon mode of wavevector q, and τ γ (τ φ ) is a typical collision time of thermal γ-quasiparticles (thermal phonons). An exten-sion to the hydrodynamic regime ω q τ γ 1 or ω q τ φ 1 may be obtained from kinetic equations [START_REF] Khalatnikov | Relaxation phenomena in superfluid Helium[END_REF]. An experimental test of our results seems nowadays at hand, either in liquid helium 4, extending the recent work of Ref. [START_REF] Fåk | Roton-phonon interaction in superfluid 4 He[END_REF], or in homogeneous cold Fermi gases, which the breakthrough of flat-bottom traps [START_REF] Gaunt | Bose-Einstein condensation of atoms in a uniform potential[END_REF] allows one to prepare [START_REF] Mukherjee | Homogeneous Atomic Fermi Gases[END_REF] and to acoustically excite by spatio-temporally modulated laser-induced optical potentials [START_REF] Kurkjian | Landau-Khalatnikov phonon damping in strongly interacting Fermi gases[END_REF]7]. Landau-Khalatnikov revisited -We recall the reasoning of Ref. [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF] to get the phonon-roton coupling in liquid helium 4, extending it to the phonon-fermionic quasiparticle coupling in unpolarised spin-1/2 Fermi gases. We first treat in first quantisation the case of a single roton or fermionic excitation, considered as a γ-quasiparticle of position r, momentum p and spin s = 0 or s = 1/2. In a homogeneous superfluid of density ρ, its Hamiltonian is given by ǫ(p, ρ), an isotropic function of p such that p → ǫ(p, ρ) is the γ-quasiparticle dispersion relation. In presence of acoustic waves (phonons), the superfluid acquires position-dependent density ρ(r) and velocity v(r). For a phonon wavelength large compared to the γ-quasiparticle coherence length [START_REF] Cohen-Tannoudji | Atomic motion in laser light[END_REF], here its thermal wavelength (2π 2 /m * k B T ) 1/2 [START_REF]For a thermal phonon wavenumber[END_REF], and for a phonon angular frequency small compared to the γ-quasiparticle "internal" energy ∆, we can write the γ-quasiparticle Hamiltonian in the local density approximation [START_REF] Thomas | The calculation of atomic fields[END_REF][START_REF] Fermi | Un metodo statistico per la determinazione di alcune prioprietà dell'atomo[END_REF]:

H = ǫ(p, ρ(r)) + p • v(r) (1)
The last term is a Doppler effect reflecting the energy difference in the lab frame and in the frame moving with the superfluid. For a weak phononic perturbation of the superfluid, we expand the Hamiltonian to second order in density fluctuations δρ(r) = ρ(r) -ρ:

H ≃ ǫ(p, ρ) + ∂ ρ ǫ(p, ρ)δρ(r) + p • v(r) + 1 2 ∂ 2 ρ ǫ(p, ρ)δρ 2 (r) (2
) not paying attention yet to the noncommutation of r and p. Phonons are bosonic quasiparticles connected to the expansion of δρ(r) and v(r) on eigenmodes of the quantum-hydrodynamic equations linearised around the homogeneous solution at rest in the quantisation volume V:

δρ(r) v(r) = 1 V 1/2 q =0 ρ q v q bq + ρ q -v q b † -q e iq•r (3)
with modal amplitudes ρ q = [ ρq/(2mc)] 1/2 and v q = [ c/(2mρq)] 1/2 q, m being the mass of a superfluid particle and c the sound velocity. The annihilation and creation operators bq and b † q of a phonon of wavevector q and energy ω q = cq obey usual commutation relations [ bq , b †

q ′ ] = δ q,q ′ . For an arbitrary number of γ-quasiparticles, we switch to second quantisation and rewrite Eq.( 2) as

Ĥ = k,σ ǫ k γ † kσ γkσ + k,k ′ ,q,σ A 1 (k, q; k ′ ) V 1/2 (γ † k ′ σ γkσ bq +h.c.) × δ k+q,k ′ + k,k ′ ,q,q ′ ,σ A 2 (k, q; k ′ , q ′ ) V γ † k ′ σ γkσ δ k+q,k ′ +q ′ × [ b † q ′ bq + 1 2 ( b-q ′ bq + h.c.)] ( 4
)
where γkσ and γ † kσ are bosonic (rotons, s = 0, σ = 0) or fermionic (s = 1/2, σ =↑, ↓) annihilation and creation operators of a γ-quasiparticle of wavevector k = p/ in spin component σ, obeying usual commutation or anticommutation relations. The first sum in the right-hand side of Eq.( 4) gives the γ-quasiparticle energy in the unperturbed superfluid, with ǫ k ≡ ǫ( k, ρ). The second sum, originating from the Doppler term and the term linear in δρ in Eq.(2), describes absorption or emission of a phonon by a γ-quasiparticle, characterised by the amplitude

A 1 (k, q; k ′ ) = ρ q ∂ ρ ǫ k + ∂ ρ ǫ k ′ 2 + v q • k + k ′ 2 (5)
where q, k and k ′ are the wavevectors of the incoming phonon and the incoming and outgoing γ-quasiparticles. Eq.( 5) is invariant under exchange of k and k ′ . This results from symmetrisation of the various terms, in the form [f (p)e iq•r + e iq•r f (p)]/2 with r and p canonically conjugated operators, ensuring that the correct form of Eq.( 2) is hermitian. The third sum in Eq.( 4), originating from the terms quadratic in δρ in Eq.( 2), describes direct scattering of a phonon on a γ-quasiparticle, with the symmetrised amplitude

A 2 (k, q; k ′ , q ′ ) = ρ q ρ q ′ ∂ 2 ρ ǫ k + ∂ 2 ρ ǫ k ′ 2 (6)
where the primed wavevectors are the ones of emerging quasiparticles. It also describes negligible two-phonon absorption and emission. The effective amplitude for φ -γ scattering is obtained by adding the contributions of the direct process (terms of Ĥ quadratic in b), and of the absorption-emission or emission-absorption process (terms linear in b) treated to second order in perturbation theory [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF]:

A eff 2 (k, q; k ′ , q ′ ) = q q ′ k k ′ + q k k ′ q ′ + q k k ′ q ′ = A 2 (k, q; k ′ , q ′ ) + A 1 (k, q; k + q)A 1 (k ′ , q ′ ; k ′ + q ′ ) ω q + ǫ k -ǫ k+q + A 1 (k -q ′ , q ′ ; k)A 1 (k -q ′ , q; k ′ ) ǫ k -ω q ′ -ǫ k-q ′ (7)
where in the second (third) term the γ-quasiparticle first absorbs phonon q (emits phonon q ′ ) then emits phonon q ′ (absorbs phonon q). Up to this point this agrees with Ref. [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF], except that the first derivative ∂ ρ ∆ in Eq.( 5), thought to be anomalously small in low-pressure helium, was neglected in Ref. [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF].

Eq. ( 7), issued from a local density approximation, holds to leading order in a low-energy limit. We then take the T → 0 limit with scaling laws

q ≈ T, k -k 0 ≈ T 1/2 (8)
reflecting the fact that the thermal energy of a phonon is cq ≈ k B T and the effective kinetic energy of a γquasiparticle, that admits the expansion

ǫ k -∆ = k→k0 2 (k -k 0 ) 2 2m * + O(k -k 0 ) 3 (9)
is also ≈ k B T . The coupling amplitudes A 1 and energy denominators in Eq.( 7) must be expanded up to relative corrections of order T [START_REF]One expands to order T 3/2 for A1 and T 2 for energy denominators[END_REF]. On the contrary, it suffices to expand A 2 to leading order T in temperature. We hence get our main result, the effective coupling amplitude of the φ -γ scattering to leading order in temperature:

A eff 2 (k, q; k ′ , q ′ ) ∼ T →0 q mcρ 1 2 ρ 2 ∆ ′′ + ( ρk ′ 0 ) 2 2m * + 2 k 2 0 2m * × ρ∆ ′ ck 0 2 uu ′ + ρ∆ ′ ck 0 (u + u ′ ) uu ′ - ρk ′ 0 k 0 + 2m * c k 0 w + m * c k 0 (u + u ′ )w + u 2 u ′2 - ρk ′ 0 k 0 (u 2 + u ′2 ) (10)
Here ∆ ′ , k ′ 0 , ∆ ′′ are first and second derivatives of ∆ and k 0 with respect to ρ; u = q•k qk , u ′ = q ′ •k q ′ k , w = q•q ′ qq ′ are cosines of the angles between k, q and q ′ ; our results hold for k 0 = 0 provided the limit k 0 → 0 is taken in Eq. [START_REF] Fermi | Un metodo statistico per la determinazione di alcune prioprietà dell'atomo[END_REF]. In Eq.(3.17) of Ref. [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF], the ∆ ′ terms were neglected as said, but the last term in Eq.( 10), with the factor ρk ′ 0 /k 0 , was simply forgotten.

Addition in the corrected-augmented version: Expression (7) of the coupling amplitude is incomplete because it does not take into account the interactions among phonons. This omission affects expression [START_REF] Fermi | Un metodo statistico per la determinazione di alcune prioprietà dell'atomo[END_REF] of the effective amplitude, the angular integral in note [47] and the dashed lines of Figs. 1 and2. The erratum [START_REF] Castin | Erratum: Landau Phonon-Roton Theory Revisited for Superfluid 4 He and Fermi Gases [PRL 119,260402[END_REF] that corrects this omission is reproduced here in appendix, see in particular Eqs. [START_REF] Berberich | Investigation of the lifetime of longitudinal phonons at GHz frequencies in liquid and solid 4 He[END_REF] and [START_REF] Rugar | Accurate measurement of lowenergy phonon dispersion in liquid 4 He[END_REF], and it is supplemented by a verification of the final result using a microscopic approach based on Bogoliubov theory with an arbitrary short-range interaction potential. -End of addition.

Damping rates -A straightforward application of Eq.( 10) is a Fermi-golden-rule calculation of the damping rate Γ scat q of phonons q due to scattering on γquasiparticles. The γ-quasiparticles are in thermal equilibrium with Bose or Fermi mean occupation numbers nγ,k = [exp(ǫ k /k B T ) -(-1) 2s ] -1 . So are phonons in modes q ′ = q, with Bose occupation numbers nb,q ′ = [exp( ω q ′ /k B T ) -1] -1 ; mode q is initially excited (e.g. by a sound wave) with an arbitrary number n b,q of phonons. By including both loss q + k → q ′ + k ′ and gain q ′ + k ′ → q + k processes [46] and summing over σ, one finds that d dt n b,q = -Γ scat q (n b,q -nb,q ) with

Γ scat q = 2π (2s + 1) d 3 kd 3 q ′ (2π) 6 A eff 2 (k, q; k ′ , q ′ ) 2 × δ(ǫ k + ω q -ǫ k ′ -ω q ′ ) nb,q ′ nγ,k ′ [1 + (-1) 2s nγ,k ] nb,q (11)
and k ′ = k + qq ′ . As our low-energy theory only holds for k B T ≪ ∆, the gas of γ-quasiparticles is nondegenerate, and nγ,k ≃ exp(-ǫ k /k B T ) ≪ 1 in Eq. [START_REF] Beliaev | Energy-Spectrum of a Non-ideal Bose Gas[END_REF]. By taking the T → 0 limit at fixed cq/k B T and setting A eff 2 = ωq ρ f , where the dimensionless quantity f only depends on angle cosines, we obtain the equivalent

Γ scat q ∼ T →0 (2s + 1) e -∆/kB T (2π) 9/2 k 2 0 q 4 c ρ 2 (m * k B T ) 1/2 I (12) with I = d 2 Ω k d 2 Ω q ′ f 2 (u, u ′ , w) an integral over solid angles of direction k and q ′ [47].
One proceeds similarly for the calculation of the damping rate Γ a-e q of phonons q due to absorption q + k → k ′ or emission k ′ → q + k processes by thermal equilibrium γ-quasiparticles. We obtain Γ a-e q = 2π (2s + 1) [START_REF] Giorgini | Damping in dilute Bose gases: A mean-field approach[END_REF] with k ′ = k + q. Low degeneracy of the γ-quasiparticles and energy conservation allow us to write nγ,k -nγ,k ′ ≃ exp(-ǫ k /k B T )/(1 + nb,q ). Energy conservation leads here to a scaling on k different from Eq.( 8) as it forces k to be at a nonzero distance from k 0 , even in the lowphonon-energy limit: When q → 0 at fixed k, the Dirac delta in Eq.( 13) becomes

d 3 k (2π) 3 [A 1 (k, q; k ′ )] 2 × δ( ω q + ǫ k -ǫ k ′ )(n γ,k -nγ,k ′ )
δ( ω q + ǫ k -ǫ k ′ ) ∼ q→0 ( cq) -1 δ 1 -u dǫ k dk c ( 14
)
and imposes that the group velocity 1 dǫ k dk of the incoming γ-quasiparticle is larger in absolute value than that, c, of the phonons. This condition, reminiscent of Landau's criterion, restricts wavenumber k to a domain D not containing k 0 . In the low-q limit, that is for q much smaller than the k significantly contributing to Eq.( 13), but with no constraint on the ratio cq/k B T , we write A 1 in Eq.( 5) to leading order q 1/2 in q, and integrate over the direction of k, to obtain

Γ a-e q ≃ (2s + 1)ρ 4πmc D dkk 2 | dǫ k dk | e -ǫ k /kB T 1 + nb,q ∂ ρ ǫ k + 2 c 2 k ρ dǫ k dk 2 (15)
∼ T →0 (2s + 1)ρk 2 * 4π 2 mc 3 ∂ ρ ǫ k * + ck * ρη * 2 k B T e -ǫ k * /kB T 1 + nb,q (16)
Eq.( 16) is an equivalent when T → 0 at fixed cq/k B T ; k * is the element of the border of D ( dǫ k dk | k=k * = η * c, η * = ±) with minimal energy ǫ k (when more than one of such k * exists, one has to sum their contributions). As ǫ k * > ∆, the damping rate due to scattering dominates the one due to absorption-emission in the mathematical limit T → 0 ; we shall see however that this is not always so for typical temperatures in current experiments.

To be complete, we give a low-temperature equivalent of the damping rate of the γ-quasiparticle k due to interaction with thermal phonons. With k -k 0 = O(T 1/2 ) as in Eq.( 8), we find Γ γφ k ∼ (πI/42)(k B T ) 7 /( cρ 1/3 ) 6 , where the factor 2s + 1 is gone (no summation over σ is needed) but I is the same angular integral as in Eq.( 12). Here scattering dominates [48]. Using τ γ ≃ 1/Γ γφ k , we checked that the figures 1 and 2 below are in the collisionless regime ω q τ γ ≫ 1. Similarly, we checked that ω q τ φ ≫ 1 on the figures. Application to helium -Precise measurements of the equation of state (relating ρ to pressure) and of the roton dispersion relation for various pressures were performed in liquid 4 He at low temperature (k B T ≪ mc 2 , ∆). They give access to the parameters k 0 , ∆, their derivatives and m * . The measured sound velocities agree with the thermodynamic relation mc 2 = ρ dµ dρ , where µ is the zerotemperature chemical potential of the liquid. We plot in Fig. 1 the phonon damping rates as functions of temperature, for a fixed angular frequency ω q . At the chosen high pressure, the phonon dispersion relation is concave at low q, therefore the Beliaev-Landau [START_REF] Beliaev | Energy-Spectrum of a Non-ideal Bose Gas[END_REF][START_REF] Pitaevskii | Landau damping in dilute Bose gases[END_REF][START_REF] Giorgini | Damping in dilute Bose gases: A mean-field approach[END_REF][START_REF] Abraham | Sound Propagation in Liquid 4 He[END_REF][START_REF] Katz | Beliaev Damping of Quasiparticles in a Bose-Einstein Condensate[END_REF][START_REF] Kurkjian | Three-phonon and four phonon interaction processes in a pair-condensed Fermi gas[END_REF] threephonon process φ ↔ φφ is energetically forbidden at low temperature and the Landau-Khalatnikov [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF][START_REF] Kurkjian | Landau-Khalatnikov phonon damping in strongly interacting Fermi gases[END_REF][START_REF] Kurkjian | Three-phonon and four phonon interaction processes in a pair-condensed Fermi gas[END_REF] pro- FIG. 1: Phonon damping rates at angular frequency ωq = 2π × 165 GHz (q = 0.3 Å-1 ) in liquid 4 He at pressure P = 20 bar as functions of temperature. Solid line: purely phononic damping Γ φφ due to Landau-Khalatnikov four-phonon processes [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF][START_REF] Kurkjian | Landau-Khalatnikov phonon damping in strongly interacting Fermi gases[END_REF][START_REF] Kurkjian | Three-phonon and four phonon interaction processes in a pair-condensed Fermi gas[END_REF]; it depends on the curvature parameter γ defined as ωq = cq[1 + γ 8 ( q mc ) 2 + O(q 4 )]. Interpolating measurements of P → γ(P ) in Refs. [START_REF] Rugar | Accurate measurement of lowenergy phonon dispersion in liquid 4 He[END_REF][START_REF] Swenson | Phonon dispersion in liquid Helium under pressure[END_REF] gives γ = -6.9. Dashed black line/dash-dotted black line: damping due to scattering/absorption-emission by rotons, see Eq.( 12)/(15). Red dashed line: original formula of Ref. [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF] for the damping rate due to phonon-roton scattering. The roton parameters are extracted from their dispersion relation k → ǫ k measured at various pressures [START_REF] Gibbs | The collective excitations of normal and superfluid 4 He: the dependence on pressure and temperature[END_REF]: ∆/kB = 7.44K, k0 = 2.05 Å-1 , m * /m = 0.11, ρk ′ 0 /k0 = 0.39, ρ∆ ′ /∆ = -1.64, ρ 2 ∆ ′′ /∆ = -8.03, ρm ′ * /m * = -4.7. In Eq.( 15), parabolic approximation ( 9) is used (hence ǫ k * /∆ ≃ 1.43). The speed of sound c = 346.6 m/s, and the Grüneisen parameter d ln c d ln ρ = 2.274 entering in Γ φφ , are taken from equation of state (A1) of Ref. [START_REF] Maris | Thermodynamic properties of superfluid 4 He at negative pressure[END_REF]. The low values q mc = 0.13 and k B T mc 2 < 10 -2 justify our use of quantum hydrodynamics.

cess φφ ↔ φφ is dominant. Our high yet experimentally accessible [START_REF] Lockerbie | Measurement of the group velocity of 93 GHz phonons in liquid 4 He[END_REF][START_REF] Dietsche | Superconducting Al-PbBi tunnel junction as a phonon spectrometer[END_REF] value of ω q leads to attenuation lengths 2c/Γ q short enough to be measured in centimetric cells. As visible on Fig. 1, the damping of sound is in fact dominated by four-phonon Landau-Khalatnikov processes up to a temperature T ≃ 0.6 K. In this regime one would directly observe this phonon-phonon damping mechanism, which would be a premiere. The sound attenuation measurements of Ref. [START_REF] Berberich | Investigation of the lifetime of longitudinal phonons at GHz frequencies in liquid and solid 4 He[END_REF] in helium at 23 bars and ω q = 2π × 1.1 GHz are indeed limited to T > 0.8 K where damping is still dominated by the rotons. Application to fermions -In cold-atom Fermi gases, interactions occur in s-wave between opposite-spin atoms. Of negligible range, they are characterized by the scattering length a tunable by Feshbach resonance [START_REF] O'hara | Observation of a strongly interacting degenerate Fermi gas of atoms[END_REF][START_REF] Bourdel | Measurement of the interaction energy near a Feshbach resonance in a 6 Li Fermi gas[END_REF][START_REF] Bartenstein | Collective excitations of a degenerate gas at the BEC-BCS crossover[END_REF][START_REF] Zwierlein | Condensation of pairs of fermionic atoms near a Feshbach resonance[END_REF][START_REF] Nascimbène | Exploring the thermodynamics of a universal Fermi gas[END_REF][START_REF] Ku | Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas[END_REF].

Precise measurements of the fermionic excitation parameters k 0 and ∆ were performed at unitarity a -1 = 0 [START_REF] Schirotzek | Determination of the superfluid gap in atomic Fermi gases by quasiparticle spectroscopy[END_REF]. Due to the unitary-gas scale invariance [START_REF] Ho | Universal thermodynamics of degenerate quantum gases in the unitarity limit[END_REF][START_REF] Enss | Viscosity and scale invariance in the unitary Fermi gas[END_REF][START_REF] Castin | The Unitary Gas and its Symmetry Properties[END_REF], k 0 is proportional to the Fermi wavenumber k F = (3π 2 ρ) 1/3 , k 0 ≃ 0.92k F [START_REF] Schirotzek | Determination of the superfluid gap in atomic Fermi gases by quasiparticle spectroscopy[END_REF], and ∆ is proportional to the Fermi energy ǫ F = 2 k 2 F 2m , ∆ ≃ 0.44ǫ F [START_REF] Schirotzek | Determination of the superfluid gap in atomic Fermi gases by quasiparticle spectroscopy[END_REF]. This also determines their derivatives with respect to ρ. Similarly, the equation of state measured at T = 0 is simply

m = ∆ 2µ , ρµ ′ /µ ≃ 0.602, ρ∆ ′ /∆ ≃ 0.815, ρ 2 ∆ ′′ /∆ ≃ -0.209, d ln c
d ln ρ ≃ 0.303). In both cases the curvature parameter γ defined in the caption of Fig. 1 is estimated in the RPA [START_REF] Kurkjian | Concavity of the collective excitation branch of a Fermi gas in the BEC-BCS crossover[END_REF]. Solid line: phononphonon (a) Beliaev-Landau damping φ ↔ φφ (for γ > 0) as in Eqs.(121,122) of Ref. [START_REF] Kurkjian | Three-phonon and four phonon interaction processes in a pair-condensed Fermi gas[END_REF] (independent of |γ|) and (b) Landau-Khalatnikov damping φφ ↔ φφ (for γ ≃ -0.30 < 0) [START_REF] Kurkjian | Landau-Khalatnikov phonon damping in strongly interacting Fermi gases[END_REF][START_REF] Kurkjian | Three-phonon and four phonon interaction processes in a pair-condensed Fermi gas[END_REF]. Dashed line/dash-dotted line: scattering/absorption-emission phonon-fermionic quasiparticle processes, as in Eq.( 12)/(15). In Eq.( 15), we took for ǫ k (a) the form proposed in Ref. [START_REF] Nishida | ǫ Expansion for a Fermi gas at infinite scattering length[END_REF] (hence ǫ k * /∆ ≃ 1.12) and (b) the BCS form (hence ǫ k * /∆ ≃ 1.14). µ is the T = 0 gas chemical potential, and the plotted quantities are in fact inverse quality factors. Here kBT /mc 2 > 0.03 in contrast to Fig. 1 where kBT /mc 2 < 0.01: cold atoms are effectively farther from the T → 0 limit than liquid helium, hence the inversion of the Γ scat q -Γ a-e q hierarchy. µ = ξǫ F , where ξ ≃ 0.376 [START_REF] Ku | Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas[END_REF], and the critical temperature is T c ≃ 0.167ǫ F /k B [START_REF] Ku | Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas[END_REF]. For the effective mass of the fermionic excitations and their dispersion relation at non vanishing k -k 0 , we must rely on results of a dimensional ǫ = 4 -d expansion, m * /m ≃ 0.56 and [START_REF] Nishida | ǫ Expansion for a Fermi gas at infinite scattering length[END_REF]. We also trust Anderson's RPA prediction [START_REF] Anderson | Random-phase approximation in the theory of superconductivity[END_REF][START_REF] Combescot | Collective mode of homogeneous superfluid Fermi gases in the BEC-BCS crossover[END_REF] that the q = 0 third derivative of the phononic dispersion relation is positive [START_REF] Kurkjian | Concavity of the collective excitation branch of a Fermi gas in the BEC-BCS crossover[END_REF]. The damping rates of phonons with wavenumber q = mc/2 are plotted in Fig. 2a. The contribution of the three-phonon Landau-Beliaev processes φ ↔ φφ, here energetically allowed, is dominant; it is computed in the quantum-hydrodynamic approximation where it is independent of the aforementioned third derivative.

ǫ k ≃ ∆ + 2 (k 2 -k 2 0 ) 2 8m * k 2 0
The phononic excitation branch becomes concave in the BCS limit k F a → 0 - [START_REF] Marini | Evolution from BCS superconductivity to Bose condensation: analytic results for the crossover in three dimensions[END_REF]. As visible on Fig. 2b, the phonon-phonon damping (now governed by the Landau-Khalatnikov processes mentioned earlier) is much weaker, and dominates the φ -γ damping only at very low temperatures. At commonly reached temperatures T > 0.05ǫ F /k B [START_REF] Hadzibabic | Fiftyfold Improvement in the Number of Quantum Degenerate Fermionic Atoms[END_REF], the damping is in fact dominated by absorption-emission φ -γ processes which, unlike in liquid helium, prevail over scattering ones because of the smaller value of ǫ k * /∆. Although the associated quality factors ω q /Γ q may seem impressive, the lifetimes Γ -1 q of the modes do not exceed one second in a gas of 6 Li with a typical Fermi temperature T F = 1µK, which is shorter than what was observed in a Bose-Einstein condensate [START_REF] Chevy | Transverse Breathing Mode of an Elongated Bose-Einstein Condensate[END_REF]. Our predictions, less quantitative than on Fig. 2a, are based on the BCS approximation for the equation of state and the fermionic excitation dispersion relation

ǫ k ≃ ǫ BCS k = [( 2 k 2 2m -µ) 2 + ∆ 2
BCS ] 1/2 and on the RPA for the q = 0 third derivative of ω q (whose precise value matters here). A cutting remark on Ref. [START_REF] Zhang | Finite-temperature damping of collective modes of a BCS-BEC crossover superfluid[END_REF]: even in the BCS approximation to which it is restricted, we disagree with its expression of Γ a-e q . Conclusion -By complementing the local density approximation in Ref. [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF] with a systematic low-temperature expansion, we derived the definitive leading order expression of the phonon-roton coupling in liquid helium and we generalized it to the phonon-pair-breaking excitation coupling in Fermi gases. The ever-improving experimental technics in these systems give access to the microscopic parameters determining the coupling and allow for a verification in the near future. Our result also clarifies the regime of temperature and interaction strength in which the purely phononic φφ ↔ φφ Landau-Khalatnikov sound damping in a superfluid, unobserved to this day, is dominant.

This project received funding from the FWO and the EU H2020 program under the MSC Grant Agreement No. 665501.

Addition in the corrected-augmented version

Erratum : corrections due to three-body interactions among phonons.

The results on the effective φ -γ scattering amplitude and associated damping rate, based on the roton Hamiltonian (Eqs. ( 1) and ( 4)), do not account for the phonon-phonon interaction, which is described, to lowest order in q, by the Hamiltonian [START_REF] Landau | Teoriya vyazkosti Geliya-II[END_REF]:

Ĥ(3) φ-φ = q1,q2,q3 δ q1+q2,q3 A 2↔1 φ-φ (q 1 , q 2 ; q 3 ) V 1/2 × b † q1 b † q2 bq3 + h.c. ( 17
)
with the φ + φ ↔ φ amplitude [START_REF] Kurkjian | Three-phonon and four phonon interaction processes in a pair-condensed Fermi gas[END_REF]:

A 2↔1 φ-φ (q 1 , q 2 ; q 3 ) = mc 2 ρ 1/2 3 q 1 q 2 q 3 32m 3 c 3 × 2 d ln c d ln ρ -1 + q 1 • q 2 q 1 q 2 + q 1 • q 3 q 1 q 3 + q 2 • q 3 q 2 q 3 (18)
We have used here the Grüneisen parameter d ln c d ln ρ = (ρµ ′′ /µ ′ + 1)/2. To leading order, there are two diagrams mediated by Ĥ(3) φ-φ missing in Eq. ( 7):

A eff 2 (k, q; k ′ , q ′ ) = q q ′ k k ′ + q k k ′ q ′ + q k k ′ q ′ + q q ′ k k ′ + q q ′ k k ′ = A 2 (k, q; k ′ , q ′ ) + A 1 (k, q; k + q)A 1 (k ′ , q ′ ; k ′ + q ′ ) ω q + ǫ k -ǫ k+q + A 1 (k -q ′ , q ′ ; k)A 1 (k -q ′ , q; k ′ ) ǫ k -ω q ′ -ǫ k-q ′ + 2A 2↔1 φ-φ (q ′ , q -q ′ ; q)A 1 (k, q -q ′ ; k ′ ) (ω q -ω q-q ′ -ω q ′ ) + 2A 2↔1 φ-φ (q, q ′ -q; q ′ )A 1 (k ′ , q ′ -q; k) ǫ k -ǫ k ′ -ω q-q ′ (19)
The energy denominators of the two new diagrams are equal to -ω q-q ′ to leading order (see note [START_REF]One expands to order T 3/2 for A1 and T 2 for energy denominators[END_REF]). One should thus add -( q/mcρ)(ρ∆ ′ /2)(2d ln c/d ln ρ -1 + w) to Eq. ( 10):

A eff 2 (k, q; k ′ , q ′ ) ∼ T →0 q mcρ 1 2 ρ 2 ∆ ′′ - ρ∆ ′ 2 2 d ln c d ln ρ -1 + ( ρk ′ 0 ) 2 2m * + 2 k 2 0 2m * × ρ∆ ′ ck 0 2 uu ′ + ρ∆ ′ ck 0 (u + u ′ ) uu ′ - ρk ′ 0 k 0 + m * c k 0 w + m * c k 0 (u + u ′ )w + u 2 u ′2 - ρk ′ 0 k 0 (u 2 + u ′2 ) (20)
This also changes the angular integral given in note [47]:

I/(4π) 2 = ( 2 k 2 0 2m * mc 2 ) 2 [ 1 25 -4α 15 + 28 45 α 2 + 2β 2 9 + A( 2 9 -4α 3 ) + A 2 + 4βB( 1 15 -α 9 ) + B 2 ( 2 15 -4α 9 + 2α 2 3 + β 2 3 ) + 2β 9 B 3 + B 4
9 ], and the definition of

A = m * ( k0) 2 [ρ 2 ∆ ′′ + ρ∆ ′ (1 - 2d ln c/d ln ρ)]+α 2 .
Accordingly, the black dashed curves in Figs. 1, 2a and 2b are multiplied respectively by 0.46, 0.78 and 0.85. Note that the Grüneisen parameter also appears in Ref. [START_REF] Nicolis | Mutual interactions of phonons, rotons, and gravity[END_REF]; our coefficients of the uu ′ and w monomials of A eff 2 are however still in disagreement with that reference.

Microscopic verification of the effective coupling amplitude

To check the expression of the effective coupling amplitude that we obtained through phonon-roton hydrodynamics (Eq. ( 20)), we recompute it within a microscopic approach in the particular case of a weakly-interacting Bose gas. In this well-known system, the excitation spectrum takes the Bogoliubov form

ǫ k = 2 k 2 2m 2 k 2 2m + 2ρV k ( 21
)
where V k is the Fourier transform of the interaction potential. This spectrum describes hydrodynamic phonons (with ǫ q = cq + O(q 3 )) provided V 0 > 0 and Next, we use the 3-and 4-quasiparticle coupling amplitudes derived from Bogoliubov theory (see Eqs. (E18), (E19) and (E20) of Ref. [START_REF] Kurkjian | Three-phonon and four phonon interaction processes in a pair-condensed Fermi gas[END_REF]), we compute the effective scattering amplitude in second order perturbation theory as prescribed by Eq. ( 105) of [START_REF] Kurkjian | Three-phonon and four phonon interaction processes in a pair-condensed Fermi gas[END_REF] (in which we set q 1 = q, q 2 = k, q 3 = q ′ and q 4 = k ′ ) and we take the limit of q and q ′ tending to 0. Since the expected result [START_REF] Rugar | Accurate measurement of lowenergy phonon dispersion in liquid 4 He[END_REF] does not depend on k to leading order, we choose k = k 0 , which greatly simplifies the microscopic calculation. As in the hydrodynamic approach, we then eliminate k ′ using momentum conservation, and the norm of q ′ using energy conservation (with the difference that one should go up to order q 3 in the calculation of q -q ′ because the scattering amplitude of the microscopic model diverges as 1/q off-shell). We get finally:

∂ k V k | k=0 = 0.
A eff micro (k, q; k ′ , q ′ ) ∼ T →0 q 16∆ 3 mcρ -2E 4 k0 + 2∆ 5 m * c 2 uu ′ + ∆E 3 k0 k 0 mc 4(u + u ′ ) + k 0 m * c uu ′ -4E 2 k0 ∆ 2 4(u 2 + u ′2 ) + w -1 + k 0 m * c uu ′ (u + u ′ ) + 2∆ 4 2w -1 + 2 k 0 m * c uu ′ (u + u ′ ) -2∆ 3 E k0 k 0 mc 2(u + u ′ ) + k 0 m * c uu ′ + 8∆ 3 mc 2 k 0 mc w(u + u ′ ) + k 0 m * c u 2 u ′2 (22)
where E k0 = 2 k 2 0 /2m. This result coincides with the hydrodynamic expression (20) specialized using the Bogoliubov dispersion relation [START_REF] Swenson | Phonon dispersion in liquid Helium under pressure[END_REF] to express k ′ 0 , ∆ ′ and ∆ ′′ in terms of k 0 , ∆ and m * :

k ′ 0 = m * k 0 E k0 mρ∆ , ∆ ′ = ∆ 2 -E 2 k0 2ρ∆ , ∆ ′′ = - 2 k 0 k ′ 0 E k0 mρ∆ - ∆ ′2 ∆ (23)
and the Bogoliubov equation of state µ = ρV 0 to obtain µ ′ = V 0 and µ ′′ = 0.

End of addition

FIG. 2 :

 2 FIG.2:Phonon damping rates at wavenumber q = mc/2 in unpolarized homogeneous cold-atom Fermi gases in thermodynamic limit as functions of temperature. (a) At unitarity a -1 = 0, where most parameters of the phonons and fermionic quasiparticles are measured (see text). (b) On the BCS side 1/kFa = -0.389, these parameters are estimated in BCS the-ory (µ/ǫF ≃ 0.809, ∆/µ ≃ 0.566, m * m = ∆ 2µ , ρµ ′ /µ ≃ 0.602, ρ∆ ′ /∆ ≃ 0.815, ρ 2 ∆ ′′ /∆ ≃ -0.209, d ln c d ln ρ ≃ 0.303).In both cases the curvature parameter γ defined in the caption of Fig.1is estimated in the RPA[START_REF] Kurkjian | Concavity of the collective excitation branch of a Fermi gas in the BEC-BCS crossover[END_REF]. Solid line: phononphonon (a) Beliaev-Landau damping φ ↔ φφ (for γ > 0) as in Eqs.(121,122) of Ref.[START_REF] Kurkjian | Three-phonon and four phonon interaction processes in a pair-condensed Fermi gas[END_REF] (independent of |γ|) and (b) Landau-Khalatnikov damping φφ ↔ φφ (for γ ≃ -0.30 < 0)[START_REF] Kurkjian | Landau-Khalatnikov phonon damping in strongly interacting Fermi gases[END_REF][START_REF] Kurkjian | Three-phonon and four phonon interaction processes in a pair-condensed Fermi gas[END_REF]. Dashed line/dash-dotted line: scattering/absorption-emission phonon-fermionic quasiparticle processes, as in Eq.(12)/(15). In Eq.(15), we took for ǫ k (a) the form proposed in Ref.[START_REF] Nishida | ǫ Expansion for a Fermi gas at infinite scattering length[END_REF] (hence ǫ k * /∆ ≃ 1.12) and (b) the BCS form (hence ǫ k * /∆ ≃ 1.14). µ is the T = 0 gas chemical potential, and the plotted quantities are in fact inverse quality factors. Here kBT /mc 2 > 0.03 in contrast to Fig.1where kBT /mc 2 < 0.01: cold atoms are effectively farther from the T → 0 limit than liquid helium, hence the inversion of the Γ scat q -Γ a-e

  It has a roton minimum∆ = ǫ k0 in k = k 0 provided ∂ k ǫ k | k=k0 = 0 and ∂ 2 k ǫ k | k=k0 = 2 /m * > 0,which is always possible with a careful choice of the function k → V k .

2m * c +O(T 5/2 ) with u and u ′ defined below Eq.( 10).

[46] We also use energy conservation and the relations 1 + (-1) 2s n = e ǫ/k B T n to transform the difference of the gain and loss quantum statistical factors.

[47] I/(4π) 2 = (

den by energy conservation, and absorption k + q ↔ k ′ , conserving energy only for q ≥ q * ≃ 2m * c/ , is O(e -ωq * /k B T ).


We theoretically and experimentally study the THz electromagnetic properties of an undoped-InAs slab whose permittivity is optically modified by a photo-generation process. The modulation of the permittivity is calculated by solving the ambipolar rate equation for the free carriers. Experiment results demonstrate that InAs is a promising semiconductor to manufacture fast and efficiently on-chip THz components. We show a high modulation of the THz transmission up to 100% from 0.75 to 1.1THz at very low pump fluence in the continuous wave regime. We also demonstrate a high-speed transmission modulation rate up to the MHz range with a modulated pump.

I. INTRODUCTION

erahertz electromagnetic waves with frequencies lying from 0.3 to 10 THz attract much interest owing to their potential applications in several domains ranging from medicine, telecommunication or security. Recent developments of bright terahertz (THz) sources and efficient detectors accelerate the progress of non-destructive THz systems. However, the demand for versatile THz components able to address a wide range of frequencies is high. Among the envisaged approaches, artificial structures named metasurfaces based on subwavelength electromagnetic resonators were used to control THz waves [START_REF] Fan | Dynamic electromagnetic metamaterials[END_REF]. Real-time and dynamical control have been demonstrated with photo-generated metasurfaces [START_REF] Okada | Photo-designed terahertz devices[END_REF][START_REF] Georgiou | Photo-generated THz antennas[END_REF]. Here, the metasurfaces are optically printed into a semiconductor layer. However, several issues must be bypassed in terms of modulation rate or pump irradiance before this approach might be implemented in the application domain.

II. DESCRIPTION OF THE STUDY

In this work, we theoretically study the THz electromagnetic properties of photo-generated carriers in an undoped-InAs slab as presented in Fig. 1. The modifications of the permittivity induced by an optical pump are calculated by solving the ambipolar rate equation for the photo-carriers. Photo-carrier diffusion plays a crucial role in the search of an optimal geometry [START_REF] Alvear-Cabezón | Photogenerated metasurfaces at terahertz frequencies induced by a continuous wave low-pump[END_REF]. To understand and to reveal the real impact of the photo-carrier diffusion in the InAs material, we have created a multiphysics code to calculate the electromagnetic properties of the InAs slab illuminated by a THz plane wave. We have analyzed the electromagnetic properties of the InAs slab as a function of the thickness "h", the pump fluence and the polarization (TE and TM) of the THz wave and demonstrated a high modulation of the THz radiations.

We have carried out the experimental demonstration described in Fig. 1. We have evaluated the transmission of a THz-wave through a micrometric-thick slab of undoped-InAs using a continuous-wave (CW) and modulated by an IR laser of 805nm wavelength. In the experimental set-up, the collimated THz wave radiates the InAs membrane located in the sample holder of the optic set-up, which in turns is traversed by the IR laser. The maximum power is 500mW and the spot diameter is 2mm when it is collimated. The rest of the optic set-up is composed of two THz wave systems: source and receiver. The THz source system is composed of a synthesizer which is used as a source, then its signal is multiplied in frequency by diodes and other amplifiers by a factor (x81). The THz receiver system is composed of an electric spectrum analyzer that sends its local oscillator (LO) on the head of an heterodyne detection, which is multiplied in frequency (x96) by diodes and other amplifiers in order to reach in the range 0.75THz to 1.1THz (RF signal). This signal is mixed with the signal measured on a diode which returns the frequency difference (IF frequency), that goes into the bandwidth of the electric spectrum analyzer, which displays this measurement. The measurement is calibrated in terms of system losses and frequency (factor x96). E. Alvear-Cabezón1,2* , S. Blin 2 , P. Nouvel 2 , F. Gonzalez-Posada 2 , R. Smaali 1 , R. Teissier 2 , A. Baranov 2 , T. Taliercio 2 and E. Centeno 1 . Fig. 2 illustrates the optical characterization of an intrinsic InAs slab of thickness h=3.8um illuminated. Fig. 2(A) illustrates the transmitted power (in dBm) for a frequency ranging from 0.75THz to 1.1THz with different pump irradiances. The spectrum shows that low-pump irradiance in the continuous regime of only a few W/cm 2 is enough to modulate the transmission of the THz waves up to 20dB. Fig. 2(B) presents the normalized THz transmission, where the spectrum shows that low irradiance (𝚽=6.6W/cm 2 ) in the CW pump is enough to modulate the transmission of the THz waves from 70% to 0% on a broad band frequency range. At last, Fig. 2(C) illustrates the THz transmission ratio intensity in dB between the InAs slab at different pump fluences T𝚽 and the reference measurement T0 when the laser pump is turnedoff (𝚽=0W/cm 2 ). We observed that an irradiation of only 𝚽=2.7W/cm 2 allows a modulation of the transmitted waves up to 6dB (blue color), and with an irradiance of 𝚽=4.7W/cm 2 is modulated up to 15dB (red color). Higher pumps overcome 20dB. The highest transmission modulation obtained is around 30dB with a very low-pump equal to 𝚽=8.1W/cm 2 . This is enough to modulate the transmission of the THz waves up to 100% at a frequency ranging from 0.75THz to 1.1THz.

We have also performed experiments using a modulated pump where we demonstrate a frequency modulation up to the MHz range. In this case, to perform the amplitude modulation of the laser pump, we use an acousto-optic modulator (AOM). The THz frequency was set to 925GHz, we set a pump irradiance of 𝚽=8.1W/cm 2 at the laser output and the zero-order of the AOM diffraction output is used to photo-generate the InAs slab. Fig. 3 presents the transmitted power as a function of time, showing the dynamic response of the THz modulation around 925 GHz. These results are limited by the zero-span operation of the spectrum analyzer. Present work consists in the observation of the modulation sidebands amplitudes of the transmitted power, and shows evidences of a possible cut off frequency close to 1MHz.

III. CONCLUSION

We have demonstrated that the undoped-InAs is a promising semiconductor for the development of on-chip fast and efficiently THz components. A modulated pump at an irradiance of few W/cm 2 is enough to dynamically control the THz transmission over a frequency range from 0.75 THz to 1 THz and with a high-speed modulation rate up to the MHz range.

Fig. 1 .

 1 Fig. 1. (A) Schematic of the Undoped-InAs slab of thickness "h" radiated by a THz plane wave from 0.75 to 1 THz. (B) The radiated InAs material is pumped by an IR plane wave which allows to obtain a modulation in transmission of the THz radiation.

Fig. 2 .

 2 Fig. 2. Experimental results of an irradiated intrinsic InAs slab of thickness h=3.8um radiated with a THz plane wave at a frequency ranging from 0.75THz to 1.1THz. (A) THz transmitted power in dBm. (B) Normalized THz transmission. (C) THz transmission ratio intensity (T𝚽 / T0).

Fig. 3 .

 3 Fig. 3.Transmitted power observed at the spectrum analyzer in a zero-span mode for a pump irradiance modulated using an on/off amplitude modulation at increasing frequencies.

Univ. Clermont Auvergne, CNRS, Institut Pascal, F-63000 Clermont-Ferrand, France.

IES, Univ Montpellier, UMR CNRS 5214, Montpellier, France.Low-pump irradiance to modulate THz waves driven by photo-generated carriers in an InAs slab

ACKNOWLEDGEMENTS

Authors thanks J.-F. Roux from IMEP-LAHC (Univ. of Savoie) for the loan of the acousto-optic modulator.