Amine Mezghani

Daniel Plabst

Lee A Swindlehurst

Inbar Fijalkow

Josef A Nossek

SPARSE LINEAR PRECODERS FOR MITIGATING NONLINEARITIES IN MASSIVE MIMO

Dealing with nonlinear effects of the radio-frequency (RF) chain is a key issue in the realization of very large-scale multi-antenna (MIMO) systems. Achieving the remarkable gains possible with massive MIMO requires that the signal processing algorithms systematically take into account these effects. Here, we present a computationally-efficient linear precoding method satisfying the requirements for low peak-to-average power ratio (PAPR) and low-resolution D/Aconverters (DACs). The method is based on a sparse regularization of the precoding matrix and offers advantages in terms of precoded signal PAPR as well as processing complexity. Through simulation, we find that the method substantially improves conventional linear precoders.

INTRODUCTION

The use of large-scale multiple antenna systems, known as Massive MIMO, has emerged as a leading concept for advancing wireless communications infrastructure. The design of such systems, however, comes with inherent challenges in terms of analog hardware and computational complexity. The rapidly growing pace of scale in antenna systems for communications necessitates methods to reduce the RF hardware complexity and power consumption, such as reducing the resolution of the data converters and operating the poweramplifiers with constant-envelope signals. In particular, the power amplifier (PA) typically accounts for a substantial amount of the power consumption in a base station (BS) [START_REF] Arnold | Power consumption modeling of different base station types in heterogeneous cellular networks[END_REF]. Therefore, it is desirable in terms of power efficiency to run the PA in the saturation region where most of the available PA power is radiated and less power is lost as heat.

Operating PAs in the saturation region and using lowresolution D/A-converters (DACs), however, implies high distortions and nonlinearities that are introduced to the signals, calling for new processing techniques to mitigate their effects. In [START_REF] Mollén | Waveforms for the Massive MIMO Downlink: Amplifier Efficiency, Distortion, and Performance[END_REF], the impact of power amplifier nonlinearity has been analyzed and it was shown that both single-carrier (SC) and OFDM transmission are affected similarly if conventional linear precoding is used since the peak-to-average power ratio (PAPR) increases with multi-user processing [START_REF] Jedda | On PAPR in Single-Carrier Massive MIMO Systems[END_REF]. Standard and improved linear zero-forcing and minimummean-squared error (MMSE) precoder designs for (massive) MIMO with low-resolution DACs and their performance have been studied in [START_REF] Mezghani | Transmit processing with low resolution D/A-converters[END_REF][START_REF] Usman | MMSE precoder for massive MIMO using 1-bit quantization[END_REF][START_REF] Saxena | Analysis of One-Bit Quantized Precoding for the Multiuser Massive MIMO Downlink[END_REF][START_REF] De Candido | Reconsidering Linear Transmit Signal Processing in 1-Bit Quantized Multi-User MISO Systems[END_REF][START_REF] Li | Downlink Achievable Rate Analysis in Massive MIMO Systems With One-Bit DACs[END_REF][START_REF] Saxena | Linear CE and 1-bit quantized precoding with optimized dithering[END_REF], showing satisfactory performance for small loading factors and well-conditioned channels such as i.i.d. channels. Achievable rates for multiuser MISO systems with low-resolution D/A converters were considered in [START_REF] Kakkavas | Weighted Sum Rate Maximization for Multi-User MISO Systems with Low Resolution Digital to Analog Converters[END_REF]. Nonlinear Tomlinson-Harashima Precoding has been considered in [START_REF] Mezghani | Tomlinson Harashima Precoding for MIMO Systems with Low Resolution D/A-Converters[END_REF] for low-resolution DACs, and achieves better performance than pure linear methods. Other works such as [START_REF] Jedda | Minimum BER precoding in 1-bit massive MIMO systems[END_REF][START_REF] Jacobsson | Quantized Precoding for Massive MU-MIMO[END_REF][START_REF] Jacobsson | Nonlinear 1-bit precoding for massive MU-MIMO with higher-order modulation[END_REF][START_REF] Swindlehurst | Minimum probability-of-error perturbation precoding for the one-bit massive MIMO downlink[END_REF][START_REF] Studer | PAR-Aware Large-Scale Multi-User MIMO-OFDM Downlink[END_REF][START_REF] Shao | A Framework for One-Bit and Constant-Envelope Precoding Over Multiuser Massive MISO Channels[END_REF][START_REF] Lopes | Optimal Precoding for Multiuser MIMO Systems With Phase Quantization and PSK Modulation via Branch-and-Bound[END_REF] show that nonlinear symbol-by-symbol precoding schemes outperform linear precoders under hardware impairments at the cost of an increased computational complexity.

Motivated by the simplicity of linear methods, we reconsider the problem of optimizing linear precoding techniques and present a new approach where SC transmission can still preserve its advantage in terms of PAPR compared to OFDM despite the multi-user processing. The new precoders are used in particular for mitigating the effects of constant envelope transmission as well as 1-bit DACs. To this end, a new cost functions with ℓ 1 regularization is investigated. The new class of linear precoders based on ℓ 1 matrix norm optimization significantly outperforms conventional designs for QPSK as well as higher-order modulations and allows for reduced computational complexity.

SYSTEM MODEL

Consider a downlink system with nonlinear transmitters:

ŝ = H • Q(P • s) + η, (1)
where H ∈ C K×N is the channel matrix with K singleantenna users and N transmit antenna, while P ∈ C N ×K is the linear precoder and Q(•) represent the transmit nonlinearity. The transmit symbol vector s is assumed to be either an OFDM or a single-carrier signal and the noise vector η is i.i.d. circular symmetric complex Gaussian with variance σ 2 η . One possible design criterion for P is that it be zeroforcing, i.e, H • P = I if quantization is ignored. However since usually N > K, the solution is not unique. Thus the question that arises is what is an appropriate solution for mitigating nonlinearities such us clipping at the power amplifier or one-bit quantization at the DAC. We will focus on two types of nonlinearities: (i) One-bit DAC quantization:

Q(y) = 1 √ 2 sign(Re{y}) + j √ 2 sign(Im{y}). (2)
and (ii) constant envelope transmission

Q(y) = exp (j∠y) , (3)
where the nonlinear operation is applied element-wise. In both cases, we obtain constant instantaneous power per antenna equal to one. We study in the next section the impact of OFDM or single-carrier processing on the PAPR and how standard linear multiuser precoding techniques affect those signals.

PAPR COMPARISON BETWEEN SINGLE-CARRIER AND OFDM SYSTEMS

In this section, we show that for the same power spectral density and the same performance under AWGN (with perfect orthogonal transmission), the worst-case peak power in the single-carrier (SC) systems grows only logarithmically with the spectral rolloff (steepness of the spectrum in the transmission band), whereas the worst-case peak power in OFDM grows linearly with the rolloff (i.e., proportional to the number of sub-carriers).

OFDM vs. block-wise single-carrier transmission

We compare OFDM and block-wise single-carrier modulation (also known as SC-FDMA in LTE) assuming an oversampling (interpolation) factor of two. In OFDM, a block of M data symbols (consisting in general of QAM symbols) is mapped to the lower frequency points of a 2M -point IFFT with zero-padding. In the block-wise single-carrier scheme, an additional M -point FFT is used to precode the data, resulting in a kind of data mapping in the time domain. The singlecarrier processing at the transmitter can also be regarded as a cyclic convolution of data blocks with the impulse response of a discrete rectangular filter with a window size of M out of 2M frequency points,

P k = 1 √ M for k < M 4 or 2M -k < M 4 -1 1 2 √ M for k = M 4 or 2M -k = M 4 -1, (4)
and zero elsewhere, with 0 ≤ k < 2M . The impulse response of the cyclic pulse shaping filter is given by applying an IFFT

p[n] = 1 for n = 0 sin(π n 2 ) M sin(π n 2M ) otherwise , 0 ≤ n < 2M. ( 5
)
Both systems have the same spectral density since the additional M -point FFT in SC-FDMA is a unitary transformation that does not change the second-order statistics. The resulting steepness of the spectral confinement is proportional to M . Also, in the AWGN case, both systems provide perfect orthogonality between data symbols and thus achieve the same BER performance. However, we show that these systems have significantly different behavior in terms of PAPR.

Peak-to-average power analysis

For simplicity, in this brief analysis we assume the input to be BPSK. For OFDM, the maximum peak-to-average-power is M and is achieved when all the inputs are identical. In the single-carrier case the peak power is achieved at the intermediate time instants between two consecutive symbols, when all symbols add coherently. Therefore, we get the peak power

M -1 n=0 |p[2n + 1]| 2 = M -1 n=0 sin π 2n+1 2 M sin π 2n+1 2M 2 = M -1 n=0 1 M sin π 2n+1 2M 2 ≤    1-1 4M 1 4M 1 sin(πx) dx    2 + 1 M sin( π 2M ) ≈ 4 π 2 log sin( π 8M ) 2 ≈ 4 π 2 log π 8M 2 , (6)
where the approximations hold for large M . Clearly, we see that the peak power of the single-carrier system only increases logarithmically with M , showing its superiority compared to OFDM in terms of PAPR. Standard linear multiuser precoding, however, diminishes this property due to the central limit theorem [START_REF] Jedda | On PAPR in Single-Carrier Massive MIMO Systems[END_REF]. Our goal is to preserve this advantage in massive MIMO using sparse precoding.

SPARSE PRECODING DESIGN

The most common choice for the zero-forcing precoder minimizes the Frobenius norm of P , i.e., the transmit power:

min P P 2 2,2 s.t. H • P = I, (7)
where the ℓ p,q matrix norm is defined as

A q p,q = j i |a i,j | p q/p . ( 8
)
The solution to this problem is famously given by the pseudoinverse of

H P = H H HH H -1 , (9)
which aims at minimizing the transmit power while fulfilling the zero-forcing requirement, or equivalently it maximizes the signal-to-noise ratio at the user terminals for a given power constraint. However, this solution results in a dense matrix P (even when H is sparse), leading to the fact that the distribution of P •s converges element-wise to a Gaussian distribution with high kurtosis. This happens even when s is QPSK, resulting in high distortion at the DACs or amplifiers and eventually producing an error floor at high SNR. Therefore, an alternative formulation is presented in the next section.

NEW APPROACH: FORMULATION BASED ON THE ℓ 1,2 NORM

If the input s is a QPSK signal and the desired output of the precoder P s should be constant envelope or even QPSK (in the case of one-bit DACs), then we propose to design the precoder to have sparse rows, in order to preserve the statistical properties of s as much as possible after precoding. To this end, we use the ℓ 1,2 matrix norm

min P P T 2 1,2 = i j |p i,j | 2 s.t. H • P = I. ( 10
)
This problem is convex and is solved efficiently. For constant envelope signaling, |p i,j | is the Euclidean norm. For the onebit DAC case, as it is applied separately to the inphase and quadrature components, it is appropriate to use a real valued representation of the channel and to redefine the norm as

|p i,j | 1-bit = |Re{p i,j }| + |Im{p i,j }|. (11)
Another important advantage of sparse precoding is the reduced computation for the matrix-vector multiplication.

Combined ℓ 1,2 /ℓ 2,2 regularization

As we will see in the simulations, the ℓ 2,2 formulation performs better at low SNR while the ℓ 1,2 approach performs better at high SNR. This is because the quantization effects dominate the noise only for high SNR values. This suggests the combination of both norms in a way that the ℓ 2,2 norm is effective at low SNR while the ℓ 1,2 norm becomes active at high SNR values, similar to the elastic net approach of [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF]:

min P HP -I 2 2,2 + 1 2 K • σ 2 η N P T 2 2,2 + λ P T 2 1,2 , ( 12
)
where λ is a constant corresponding to the value K•σ 2 η,cross /N at exactly the crossing point of both BER curves (c.f. Fig. 2). This combined ℓ 1,2 /ℓ 2,2 regularized optimization can be solved iteratively using the iterative shrinkage-thresholding algorithm (ISTA) [START_REF] Yang | Fast ℓ 1 -minimization algorithms and an application in robust face recognition: A review[END_REF] with an appropriate stepsize µ

P ℓ+1 = exp j∠(P ℓ -µ∆ ℓ ) • max P ℓ -µ∆ ℓ -µ λ 2 • ( k |p ℓ k |) • 1 T , 0 , (13)
where • represents the Hadamard product and

∆ ℓ = H H H + 1 2 K • σ 2 η N • I P ℓ -H H . ( 14
)
In the iterative formula [START_REF] Jacobsson | Quantized Precoding for Massive MU-MIMO[END_REF], the absolute values are taken elementwise (note that for 1-bit we define |x+yj| = |x|+|y|) and p k are the column vectors of P .

Superposition coding for higher-order modulation

Using higher-order modulation such as 16-QAM for better spectral efficiency while the base station only employs 1-bit DACs at each antenna may seem to be contradictory at first glance. However since the number of antennas is much larger than the number of users, such an idea can be achieved in principle with an appropriate transmit strategy. We propose such a method, still based on linear precoding, which is applicable for square QAM constellations. To this end, we apply the concept of superposition coding [START_REF] De Candido | Reconsidering Linear Transmit Signal Processing in 1-Bit Quantized Multi-User MISO Systems[END_REF], to reformulate the problem again based on QPSK symbols that superimpose "over-theair" through the channel H to form the desired QAM constellation at the user terminals. For simplicity, let us consider the 16-QAM case, which can be represented as the sum of two QPSK signals: One least significant symbol (LSB) and one most significant symbol (MSB)

s 16QAM = (I K ⊗ [1, 2]) • s QPSK = Π • s QPSK , (15)
where s QPSK ∈ {±1 ± j} 2K . Now, instead of designing a precoder matrix P to be applied directly to s 16QAM , we design a larger precoder P ∈ C N ×2K that is applied to s QPSK , and we let the corresponding signals superimpose through the channel and form the desired signal s 16QAM at the user terminals. By doing so and additionally encouraging P to be sparse, we can produce a binary-like signal at the output of the precoder and mitigate the issue of the 1-bit DACs, while at the receiver side a higher-order modulated signal is generated through the channel.

Unfortunately, with the new constraint HP = Π, the Restricted Isometry Property (RIP) condition is not valid for this problem meaning that the ℓ 1 solution might not be sparse as desired. In fact, due to symmetry considerations (same channel for both MSB and LSB bits), the solution will present the same symmetry, i.e., the precoding vector for the most significant bit is just twice the one for the LSB bit. To break this symmetry, we propose to add some small non-convex perturbation of the original problem in order to encourage sparse solutions having different precoding vectors for both bits:

P HP -Π 2 2,2 + 1 2 K • σ 2 η N P T 2 2,2 + λ P T 2 1,2 + tr(P JP H ) , (16)
with J = I K ⊗ (11 T -I 2 ). The term tr(P JP H ) represents the non-convex perturbation for breaking the symmetry. Again, the iterative shrinkage-thresholding algorithm can be used to solve this optimization (locally).

SIMULATION RESULTS

We consider first a system with N = 30 antennas, K = 5 users, and QPSK constellations. The channel entries are h ij ∼ CN (0, 1). In the simulations, we used a conservative value for the stepsize µ = 0.01. Accelerated ISTA versions (such as FISTA) can be also used for significantly faster convergence rate [START_REF] Yang | Fast ℓ 1 -minimization algorithms and an application in robust face recognition: A review[END_REF]. In Fig. 1, the complementary cumulative distribution function (CCDF) of the output power is shown for the new precoding technique with ℓ 1 regularization compared to the conventional ZF-technique for different modulation formats. For comparison, the CCDF without precoding is also shown. With the new precoder, the PAPR is only increased marginally, while SC-FDMA and RRC pulse shaping still preserve their advantage compared to OFDM. The BER performance is shown in Fig. 2 and Fig. 3 under constant envelope and 1-bit DAC constraints, respectively. In both cases, we observe substantial improvements at high SNR with sparse precoding. As explained earlier, the elastic net with the combined ℓ 1 and ℓ 2 regularization provides the best performance over the entire SNR range as it optimizes the trade-off between beamforming gain and PAPR reduction.

Finally, we consider 16-QAM modulation with superposition coding and 1-bit DACs for K = 8 and N = 400 in Fig. 4. At the receive side, we use the following blind estimation method for the scaling factor for each user prior to detection, as proposed in [START_REF] Jedda | Precoding under instantaneous per-antenna peak power constraint[END_REF]:

f k = T • E [|Re{s}| + |Im{s}|] T t=1 |Re{ŝ k [t]}| + |Im{ŝ k [t]}| , ( 17
)
where T is the length the received sequence. Again, sparse linear precoding combined with superposition coding is clearly advantageous at high SNR. The proposed 16-QAM precoder design also better exploits the higher antenna count than standard ZF at higher SNR values.

CONCLUSION

We introduced a novel sparse precoding method to cope with the PA and DAC issues in large-scale MIMO. We found the method to substantially improve the performance of standard linear precoders while reducing the amount of computation required to implement the linear precoder. With the increased demands for antenna element counts, we believe techniques offering reduced analog and digital hardware complexity such as the method presented here will be an essential ingredient in the development of future ultra-massive MIMO systems.

Fig. 1 .

 1 Fig. 1. OFDM vs. single-carrier in terms of the CCDF of instantaneous output power for QPSK input and M = 128, N = 30 antennas, and K = 5 users.

Fig. 2 .

 2 Fig. 2. BER Performance with constant envelope precoding for the downlink with K = 5 users and N = 30 antennas, QPSK.

Fig. 3 .BERFig. 4 .

 34 Fig. 3. BER for K = 5 and N = 30, QPSK, with 1-bit DACs.


Ming-Guang Hu

Yu Liu

Matthew A Nichols

Lingbang Zhu

Goulven Quéméner

Olivier Dulieu

Kang-Kuen Ni
email: ni@chemistry.harvard.edu

Nuclear spin conservation enables state-to-state control of ultracold molecular reactions

Quantum control of reactive systems has enabled microscopic probes of underlying interaction potentials, the opening of novel reaction pathways, and the alteration of reaction rates using quantum statistics. However, extending such control to the quantum states of reaction outcomes remains challenging. In this work, we realize this goal through the nuclear spin degree of freedom, a result which relies on the conservation of nuclear spins throughout the reaction. Using resonance-enhanced multiphoton ionization spectroscopy to investigate the products formed in bimolecular reactions between ultracold KRb molecules, we find that the system retains a near-perfect memory of the reactants nuclear spins, manifested as a strong parity preference for the rotational states of the products. We leverage this effect to alter the occupation of these product states by changing the superposition of initial nuclear spin states with an external magnetic field. The techniques demonstrated here open up the possibility to study ultracold reaction dynamics at the state-to-state level as well as quantum entanglement between reaction products.

Controlling reactions between molecules at the level of individual quantum states is a long-standing goal in chemistry and physics [1]. Recent advancements in cold and ultracold techniques granted control over all molecular degrees of freedom, including electronic, rotation, vibration, nuclear spin, and partial waves of scattering [2,3]. This has enabled the understanding and manipulation of chemical reactions free from thermal ensemble averaging [4][5][6]. In particular, the preparation of cold reactants in single quantum states has facilitated studies of scattering resonances which probe interaction potentials with exceptional resolution [7][8][9][10], efficient steering of reactive pathways to form selected product species [11,12], and modification of long-range interactions to dramatically alter the overall rates of reactions [13][14][15][16][17]. While these results have improved our fundamental understanding of chemical processes, gaining control over the quantum states in which reaction products are formed is a next milestone [18]. Despite much experimental effort, the ability to detect, let alone exert control over, the quantum states of product species in these cold and ultracold reactions has remained challenging.

A fundamental factor standing in the way of product control in low-temperature reactions is the fact that they are typically barrierless, and proceed through the formation of intermediate complexes [19], which can live for a duration significantly longer than the characteristic rovibrational timescale of the system. As a result, the energy of the complex is expected to be well-mixed into its various degrees of freedom, leading to a statistical distribution of the reaction products in which the effect of the reactant state preparation is strongly suppressed [START_REF] Levine | Molecular Reaction Dynamics[END_REF]. One notable exception to this behavior, however, is that spins of the nuclei involved tend to remain unchanged throughout reactions [START_REF] Quack | [END_REF]22]. This phenomenon has been observed in both complex-forming [23][24][25] and direct [26] bimolecular reactions, as well as in photodissociations [27][28][29], and was attributed to weak coupling between nuclear spins and other degrees of freedom. As such, nuclear spins provide a promising handle via which control over ultracold reactions can be exerted.

Previous studies on the behavior of nuclear spins have focused on reactions involving hydrogen-containing species, due largely to their relevance in astrochemical processes. The results of these studies helped establish the well-accepted rule of nuclear spin conservation in chemical reactions [22]. Ultracold reactions, on the other hand, display unique characteristics that may challenge the robustness of this rule. Specifically, nuclei involved in these reactions can possess spins many times higher than those found in prior works, which were protons with spin of 1/2. This can result in stronger couplings between the spins and other degrees of freedom. Furthermore, the intermediate complex involved in these reactions can be exceptionally long-lived [30,31], which provides more time for these couplings to alter the spins. Thus, the behavior of nuclear spins in ultracold reactions is an open question and remains to be experimentally examined. In prior studies, such examinations were carried out in systems involving cascaded reactions, using reactants with statistically mixed nuclear spins. Ultracold molecules, in contrast, can be prepared in single nuclear spin states, and react via single collisions. Hence, they provide a unique platform to cleanly test the conservation of nuclear spins.

In this study, we show that nuclear spins are conserved throughout the ultracold reaction 40 K 87 Rb+ 40 K 87 Rb → K 2 Rb * 2 → 40 K 2 + 87 Rb 2 (Fig. 1(a)), where the K and Rb nuclei have high nuclear spins of 4 and 3/2, respectively, and the intermediate complex K 2 Rb * 2 has a long lifetime of 360 (30) ns [32]. Because the products are homonuclear diatomic molecules, by the exchange symmetry of identical particles, their rotational states are correlated to their nuclear spin states [START_REF] Atkins | Molecular Quantum Mechanics[END_REF]. Using resonance-enhanced multiphoton ionization (REMPI) spectroscopy to probe the rotational states of the products, we find K 2 molecules predominantly in even-parity rotational states (N

K 2 = 0, 2, • • • , 12) and Rb 2 in odd (N Rb 2 = 1, 3, • • • , 19),
consistent with nuclear spin conservation. Utilizing this conservation, we demonstrate a technique to continuously alter the state distributions of the products by changing the superposition state of the reactant nuclear spins with an applied magnetic field. We model this behavior as a projection of the spin state of the reactant nuclei onto the symmetric or antisymmetric spin states of the product nuclei.

Our experiment begins with a trapped gas of X 1 Σ + (v = 0, N KRb = 0) ground-state KRb molecules prepared in a single hyperfine state. Here v and N are vibrational and rotational quantum numbers, respectively. In each experimental cycle, we typically prepare 10 4 molecules in a crossed optical dipole trap with a temperature of 500 nK and a peak density of 10 12 cm -3 . Details of the apparatus regarding the production and detection of the gas were reported elsewhere [START_REF] Liu | [END_REF]. In the presence of a sufficiently large magnetic bias field (B > 20 G), the spins of the K and Rb nuclei are decoupled from one another, and the hyperfine state of the molecules can be written as

|i K = 4, i Rb = 3/2, m K i = -4, m Rb i = 1/2
, where i and m i are the nuclear spins and their projections along the bias field, respectively [35,36].

Once prepared, the molecules undergo the exchange reaction

2KRb(v = 0, N KRb = 0) → 40 K 2 (v = 0, N K 2 ) + 87 Rb 2 (v = 0, N Rb 2 ), (1)
for which the product species were previously confirmed by combining single-photon ionization and ion mass spectrometry [37]. Here, using state-controlled reactants at B = 30 G, we investigate the conservation of nuclear spins throughout this reaction.

While directly probing the nuclear spin states of products is challenging due to their small energy splitting, we can infer this information from the parity of their rotational states.

For homonuclear diatomic molecules such as K 2 and Rb 2 , nuclear spin and rotation are linked by the exchange symmetry of identical particles [START_REF] Atkins | Molecular Quantum Mechanics[END_REF]. In 40 K 2 , where the nuclei are bosonic, spin states that are symmetric under the exchange of nuclei require rotational states with even parity, while those that are antisymmetric require odd parity. In 87 Rb 2 , where the nuclei are fermionic, this symmetry-parity correspondence is switched. Therefore, if nuclear spins are conserved over the reaction, K 2 and Rb 2 will have symmetric spin states

X 1 Σ + g (v = 0, NK 2 ) → B 1 Πu(v = 1, N K 2 )
|i K = 4, i K = 4, m K i = -4, m K i = -4 and |i Rb = 3/2, i Rb = 3/2, m Rb i = 1/2, m Rb i = 1/2 ,
respectively, allowing only even rotational states of K 2 and odd rotational states of Rb 2 to be occupied. This outcome is easily distinguishable from that where the nuclear spin states are statistically occupied. In this case, the population ratio between even and odd rotational states would be 5/4 for K 2 and 3/5 for Rb 2 , as determined by the numbers of symmetric and antisymmetric spin states in each molecule [START_REF] Atkins | Molecular Quantum Mechanics[END_REF].

To detect the product rotational states, we use a 1 + 1 REMPI technique, which con-sists of an initial single-photon bound-to-bound transition from the electronic and vibrational ground-state X 1 Σ + g (v = 0, N ) to an electronically excited intermediate-state B 1 Π u (v , N ), followed by a single-photon bound-to-continuum transition that ionizes the molecules (Fig. 1(b)). We drive the bound-to-bound transition using a frequency-tunable laser operating around 648 nm for the detection of K 2 and 674 nm for Rb 2 . The bound-tocontinuum transition is excited by a 532 nm pulsed laser for both product species. Figure 1(c) shows the timing diagram for product ionization and detection. Because the trap light can alter the outcome of the reaction [32], we apply a 7 kHz square wave modulation to its intensity, and probe the products during the dark phases of the modulation using REMPI pulses and ion mass spectrometry. In each cycle, ion signals are recorded until all the KRb reactants in the sample are depleted (∼ 1 s). To avoid photoexcitation of the KRb molecules by the REMPI lasers [38,39], we shape the beams to have dark spots centered on the trapped sample (Fig. S1). Products, on the other hand, escape the trap because of their significant translational energies, and a reasonable portion (∼ 30%) of them are illuminated by the REMPI beams before they leave the detection region.

To identify the occupied rotational states of the products, we scan the 648/674 nm laser frequency to search for resonances around the ∆N = N -N = 0 (Q branch) transitions, and compare our measurements with theoretical calculations based on molecular potentials fitted by prior spectroscopic data [40][41][42][43][44]. The results are shown in Fig. 2 for K 2 and Fig. 3 for Rb 2 . We observe resonant signals corresponding to the states N = 0 -12 for K 2 and 0 -19 for Rb 2 , at frequencies that match the predicted values to within our measurement precision. The spectral widths of these resonances (insets) arise from the natural linewidth, the laser frequency uncertainty, and the Doppler width (see SM). We also search for states beyond N K 2 = 12 and N Rb 2 = 19, which are energetically forbidden based on the calculated reaction exothermicity of 9.54 cm -1 [45], and find their populations to be consistent with zero. Among those that are energetically accessible, we find populations predominantly in the even states of K 2 and odd states of Rb 2 . This significantly deviates from statistical behavior, and provides direct evidence for the conservation of nuclear spins in a reaction involving high-spin nuclei and a long-lived intermediate complex. We note that the amplitudes of the resonances in Figs. 2 and3 are not yet sufficient to directly reconstruct the intrinsic product state distribution governed by the reaction dynamics, due to uncertainties in the state-dependent ionization efficiencies.

Given the strong rotational parity preference exhibited by a simple hyperfine state of the reactants, one can consider how such conservation manifests itself for a more general form of the state. To explore this, we adjust the spin superposition of the KRb reactants by tuning the bias field. After initializing the molecules in the state |4, 3/2, -4, 1/2 , we adiabatically ramp the field to a final value B, where the nuclear spin state of reactants takes the form,

ψ KRb = α 4, 3 2 , -4, 1 2 + β 4, 3 2 , -3, - 1 2 + γ 4, 3 2 , -2, - 3 2 .
(

) 2
The admixture probabilities, |α| 2 , |β| 2 , and |γ| 2 , are calculated by diagonalizing the hyperfine Hamiltonian of KRb [35,36] as a function of B, as shown in Fig. 4 loss of generality, we measure changes in the amplitudes of the resonances corresponding to the rotational states N = 6, 7 for K 2 and N = 4, 5, 13, 14 for Rb 2 (Fig. 4(c,d)). We observe that the odd (even) states of K 2 (Rb 2 ), which are unoccupied at higher magnetic fields, acquire significant population as B decreases, while the population in the even (odd) states are suppressed. This result shows that the relative occupation of odd and even parity states of the products can be altered experimentally by controlling the nuclear spins of reactants via an external field.

We model this behavior using a selection rule based on angular momentum algebra, which is derived in accordance with the conservation of nuclear spins throughout the reaction (SM). Given the form of the reactant input state (Eq. 2), the probabilities for scattering into the antisymmetric (A) and symmetric (S) spin states of K 2 or Rb 2 are found to be 2 and P S = 1 -P A , respectively. The magnetic field dependence of these probabilities, shown in Fig. 4(b), derives from that of the admixture coefficients (Fig. 4(a)). Because of the exchange symmetry of identical nuclei, the detected populations of even (odd) rotational states are proportional to P S (P A ) for K 2 and P A (P S ) for Rb 2 .

P A = |αβ| 2 + |αγ| 2 + |βγ|
The constant of proportionality depends on the rotational state distribution resulting from the reaction dynamics, as well as the state-dependent ionization efficiency, and is therefore unique for each rotational state. As a result, we fit each data set in Fig. 4(c,d) to the function a • P S/A + b, with free parameters a and b. Here, a represents the aforementioned proportionality constant, and b is an offset arising from the noise floor of the detection. The fitted curves (Fig. 4(c,d)) show that our model for P S/A describes well the magnetic field dependence of the rotational state populations.

While this model was derived for a specific input state, Eq. ( 2), the methodology does not depend on the particular form of this state, and can be generalized to account for arbitrary states of the reactant nuclear spins. Experimentally, such states may be realized using a combination of external magnetic fields and microwave control techniques [35,[46][47][48]. With this capability, the product state control demonstrated here can be used to establish selected parity correlations between products originating from the same reaction. Such correlations will allow for the generation of quantum entanglement between reaction products [49,50].

targeted transitions. The pulse energy (optical power) of the 532 nm laser is set to be 50 µJ (350 mW), resulting in the ionization of > 80% of the electronically-excited molecules. The frequencies of the diode lasers are stablized to an accuracy of 25 MHz through locking to a laser wavelength meter (Bristol 771A).

To avoid excitation of the reactant KRb molecules by the REMPI lasers, we generate dark spots in the centers of the beams using the optical setup illustrated in Fig. S1. The 648/674 nm and 532 nm beams are first combined on a dichroic mirror, and are then sent through a dark mask (Thorlabs R1DF100). The mask is located in the object plane of an achromatic lens (f = 300 mm), and is projected onto the image plane coinciding with the molecular cloud location, resulting in the beam profiles shown in the insets. The dark spot size is chosen based on the KRb cloud size, which is up to 80 µm (4σ width) in its longest direction. The length of the dark region along the direction of propagation is approximately 7 mm on either side of the image plane (defined as when the optical intensity inside the dark spot increases to half of the maximal value measured in the image plane).

For 40 K 2 , the bound-to-bound transition lines we target are from the

X 1 Σ + g (v = 0, N ) → B 1 Π u (v = 1, N ) vibronic band.
The transition dipole moment for these lines is calculated to be (1.6 e • a 0 ) S P,Q,R N . Here, e is the electron charge, a 0 is the Bohr radius, and S P,Q,R N is the Hönl-London factor which takes different forms for P , Q, and R branch rotational transitions. For Σ → Π electronic transitions, they are given by S P N = N -1 2N +1 , S Q N = 1, and S R N = N +2 2N +1 [START_REF] Bernath | Spectra of Atoms and Molecules[END_REF]. We calculate the linewidth (lifetime) of the excited state to be 14.5 MHz (11 ns), close to the measured value of 13 MHz (12.2 ns) reported in Ref. [START_REF] Lemont | [END_REF]. An additional contribution to the experimentally measured linewidth is Doppler broadening arising from the large translational energy of K 2 products. For K 2 molecules moving at the maximum velocity allowed by the reaction exothermicity, the Doppler contribution to the linewidth is σ D ∼ 40 MHz. The REMPI spectroscopy in this work requires knowing transition frequencies better than 0.001 cm -1 (or 30 MHz), which is not available from the literature. To identify them, our initial search of the transition lines is guided by frequencies that are calculated using molecular potentials from Ref. [40,41].

For 87 Rb 2 , the bound-to-bound transition lines we target are from the X 1 Σ + g (v = 0, N ) → B 1 Π u (v = 4, N ) vibronic band. We calculate the transition dipole moment to be (1.4 e • a 0 ) S P,Q,R N , and the natural linewidth (lifetime) of the excited state to be 13.9 MHz (11 ns). For Rb 2 molecules moving at the maximum velocity, the Doppler contribution to the linewidth is σ D ∼ 30 MHz. Our initial search of the transition lines is guided

H S = -µ • E, (S4) H Z = -g r µ N N • B -g K µ N (1 -σ K ) I K • B -g Rb µ N (1 -σ Rb ) I Rb • B, (S5)
where N is the rotational angular momentum operator, I K,Rb are the nuclear spin operators for the K and Rb nuclei, respectively, (∇E) K,Rb are the intramolecular electric field gradients at the K and Rb nuclei, respectively, eQ K,Rb are the nuclear electric quadrupole moment operators for K and Rb, respectively, µ is the molecular dipole moment, E is the external electric field, B is the external magnetic field, and e is the charge of the electron.

In this molecular Hamiltonian, Eq. (S2) denotes the rotational contribution to the energy, with corresponding rotational constant B rot /h = 1.1139514 GHz [35,[START_REF] Neyenhuis | [END_REF]. Eq. (S3)

represents the hyperfine energy, where -e (∇E) i • Q i describes the interaction between the intramolecular electric field gradient at nucleus i and the corresponding nuclear electric quadrupole moment, which is characterized by the electric quadrupole coupling constants (eqQ) K /h = 0.452 MHz and (eqQ) Rb /h = -1.308 MHz [START_REF] Neyenhuis | [END_REF]. The remaining three terms in Eq. (S3) describe the interactions of the individual nuclear spins with the magnetic field associated with the rotation of the molecule, where the corresponding coupling constants are given by c K /h = -24.1 Hz and c Rb /h = 420.1 Hz [36], as well as the scalar nuclear spin-spin interaction, with an associated coupling constant c 4 /h = -2030.4 Hz [36]. The Stark Hamiltonian given by Eq. (S4) describes the interaction between the permanent electric dipole moment of KRb, µ = 0.574 Debye [START_REF] Ni | A Quantum Gas of Polar Molecules[END_REF], and an external electric field. The Zeeman contribution to the energy (Eq. (S5)), on the other hand, consists of three separate terms:

the interaction between the external magnetic field and the magnetic moment arising from molecular rotation, with a corresponding rotational g-factor of g r = 0.014 [36], and the interactions between the magnetic moments of the individual nuclei, whose g-factors are

given by g K = -0.324 and g Rb = 1.834 [36], and the external magnetic field. The remaining parameters in Eq. (S5), µ N and σ K,Rb , represent the nuclear magneton and the nuclear shielding constants, respectively. For the K and Rb nuclei, the nuclear shielding constants are σ K = 1321 ppm and σ Rb = 3469 ppm [36].

For the reactant state-decomposition calculations shown in the main text (Fig. 4(a)),

we include rotational states up to N max = 1 when diagonalizing this Hamiltonian in the basis of uncoupled hyperfine states, N, m N , i K , m K i , i Rb , m Rb i , where i K = 4 and i Rb = 3/2. We also make use of the fact that the ground-state molecules are initially prepared at zero electric field and at a high magnetic field, B ≈ 544 G, where the un-

19 40K 2 2 FIG. 1 .

 19221 FIG. 1. Testing the conservation of nuclear spins in bimolecular reactions between ultracold KRb molecules using REMPI spectroscopy. (a) Schematic illustration of the conservation of nuclear spin throughout the reaction KRb + KRb → K2Rb * 2 → K2 + Rb2 at B = 30 G. In this case, the initial reactant nuclear spin states m K i = -4 and m Rb i = 1/2 are inherited by the product nuclei, resulting in symmetric product nuclear spin states. Due to the exchange symmetry of the identical bosonic (fermionic) nuclei of K2 (Rb2), their rotational states are restricted to even (odd) parity. (b) The REMPI scheme used to identify rovibrational states of products. K2 and Rb2 molecules are state-selectively photoionized from the initial quantum state, X 1 Σ + g (v = 0, N ), via an intermediate state, B 1 Πu(v , N ). Here, v and v are vibrational quantum numbers; N and N are rotational quantum numbers. (c) Timing diagram for product ionization and detection. The optical trap confining the reactant KRb molecules is square-wave modulated at 7 kHz. During the dark phase of the modulation, K2 and Rb2 products are generated by the reaction, ionized using REMPI, and detected by ion mass spectrometry. The sequence is repeated for a duration of 1 s, until all reactants in the trap are depleted.

6 FIG. 2 .

 62 FIG.2. REMPI spectrum for 40 K2 product molecules. By scanning the 648 nm laser frequency to search for rotational lines within theX 1 Σ + g (v = 0, NK 2 ) → B 1 Πu(v = 1, N K 2 )vibronic band, we observe strong K + 2 signals at frequencies corresponding to rotational states with even values of NK 2 , and highly suppressed signals for odd values.

  vibronic band, we observe strong K + 2 signals at frequencies corresponding to rotational states with even values of NK 2 , and highly suppressed signals for odd values. The ion count for each data point is normalized by the corresponding number of experimental cycles (∼ 16); the error bars denote shot noise. For NK 2 > 0 we drive transitions with N K 2 -NK 2 = 0 (Q branch), whereas for NK 2 = 0, we drive the only allowed transition, with N K 2 -NK 2 = 1 (R branch). Blue dashed lines indicate the predicted transition frequencies. The relatively strong asymmetric signal associated with NK 2 = 5 is due to a nearby NK 2 = 2 → N K 2 = 1 (P branch) transition at 462925.68 GHz. We do not observe any signals at frequencies corresponding to states with NK 2 > 12. Gaussian fits (red curves) are applied to each signal peak, yielding a typical spectral linewidth (1σ) of ∼ 50 MHz. (Inset) Lineshape for the transition NK 2 = 6 → N K 2 = 6; the Gaussian width is σ = 46 ± 4 MHz.

13 FIG. 3 .

 133 FIG.3. REMPI spectrum for 87 Rb2 product molecules. The frequency of the 674 nm laser is scanned withinthe X 1 Σ + g (v = 0, N Rb 2 ) → B 1 Πu(v = 4, NRb2 ) vibronic band. We observe strong Rb + 2 signals for transitions from odd rotational states, and highly suppressed signals from even ones. The ion count for each data point is normalized by the corresponding number of experimental cycles (∼ 16); the error bars denote shot noise. We drive Q branch transitions for N Rb 2 > 0, and R branch for N Rb 2 = 0. Blue dashed lines indicate the predicted transition frequencies. The relatively strong asymmetric signal associated with N Rb 2 = 6 is due to a nearby N Rb 2 = 15 → N Rb 2 = 16 (R branch) transition at 445032.31 GHz, while that with N Rb 2 = 2 is due to a N Rb 2 = 13 → N Rb 2 = 14 (R branch) transition at 445035.30 GHz. We do not observe any signals at frequencies corresponding to states with N Rb 2 > 19. Gaussian fits (black curves) are applied to each signal peak, yielding a typical spectral linewidth (1σ) of ∼ 40 MHz. (Inset) Lineshape for the transition N Rb 2 = 13 → N Rb 2 = 13; the Gaussian width is σ = 43 ± 3 MHz.

FIG. 4 .

 4 FIG. 4. Controlling the occupation of product rotational states using an external magnetic field. (a) Calculated admixture probabilities |α| 2 , |β| 2 , and |γ| 2 for the reactant spin superposition (Eq. 2) as a function of magnetic field. (b) Probabilities for scattering into the symmetric (PS) and antisymmetric (PA) nuclear spin states of K2 or Rb2 based on the conservation of nuclear spins. (c,d) Product ion counts associated with the rotational states N = 6, 7 for K2 (c) and N = 4, 5, 13, 14 for Rb2 (d) obtained at different magnetic field values between 5 and 50 G. For each data set, the 648/674 nm laser frequency is fixed to be resonant with the corresponding transition, as identified from the spectra in Figs. 2 and 3. The ion count for each data point is normalized by the corresponding number of experimental cycles (∼ 20). Error bars include shot noise and 10% ion number fluctuation due to the REMPI laser frequency uncertainty (∼ 25 MHz). Red (blue) curves are fits to the function a • P S(A) + b, where a and b are fit parameters representing the state-dependent proportionality constant and the offset arising from the detection noise floor, respectively. The typical root-mean-square error (RMSE) for the fit is 0.35.

ACKNOWLEDGMENTS We thank T. Rosenband for providing the code used to calculate the molecular hyperfine structure, J. Bohn for discussions, and R. Vexiau for calculations of relevant molecular transitions. Funding: This work is supported by the DOE Young Investigator Program, the David and Lucile Packard Foundation, and the NSF through the Harvard-MIT CUA. M.A.N. is supported by a HQI fellowship. G.Q. acknowledges funding from the FEW2MANY-SHIELD Project No. ANR-17-CE30-0015 from Agence Nationale de la Recherche.

[1] J. C. Polanyi, Accounts of Chemical Research 5, 161 (1972).

Supplementary Material

S1. 1+1 REMPI SPECTROSCOPY

Our REMPI scheme for the state-selective detections of K 2 (Rb 2 ) utilizes two lasers: a continuous-wave external cavity diode laser operating at 648 (674) nm, and a pulsed Nd:YVO 4 laser with a frequency-doubled output wavelength of 532 nm and a pulse duration of ∼ 10 ns. During an experiment, temporally-overlapping light pulses derived from these lasers are repeatedly applied to ionize reaction products. The duration of the 648 (674) nm pulse is controlled by an acoustic-optical modulator (AOM) to be ∼ 40 ns, and the relative delay between the two pulses is scanned to maximize the resulting ion signals. The optical power in the 648 and 674 nm beams is 8 and 15 mW, respectively, sufficient to saturate the by frequencies that are calculated using molecular potentials from Ref. [42][43][44].

S2. STATE-DECOMPOSITION CALCULATION FOR REACTANT MOLECULES

To obtain the coefficients α, β, and γ of the state-decomposition given in Eq. (2) of the main text, we diagonalize the molecular Hamiltonian for the 40 K 87 Rb reactants using the basis of uncoupled hyperfine states. If we include the rotational degrees of freedom, we can label these basis states by N, m N , i K , m K i , i Rb , m Rb i , where N represents the rotational angular momentum quantum number, m N is its projection onto the z-axis, i K,Rb are the nuclear spins of the K and Rb atoms, and m K,Rb i are the corresponding projections onto the z-axis. The direction of the z-axis here is defined by an externally applied magnetic field. In the electronic ground state, X 1 Σ + , as well as the vibrational ground state, the molecular Hamiltonian in the presence of external electric and magnetic fields can be expressed as [36,[START_REF] Brown | Rotational Spectroscopy of Diatomic Molecules[END_REF],

Here,

)

coupled basis states are a good representation of the eigenstates of the molecular Hamiltonian (Eq. (S1)). In this configuration, the KRb molecules are produced in their rotational, vibrational, and electronic ground state, and in the hyperfine state [START_REF] Liu | [END_REF]. This corresponds to the situation where α ≈ 1 and β = γ ≈ 0 in Eq. ( 2) of the main text. As the electric and magnetic fields are changed adiabatically following the initial preparation of the reactant molecules, we obtain the coefficients α, β, and γ by diagonalizing the molecular Hamiltonian (Eq. (S1))

at the desired values of the electric and magnetic fields, and selecting the eigenstate which is adiabatically connected to the initial hyperfine state, |ψ i KRb . The coefficients of the expansion of this eigenstate in terms of the uncoupled basis states are then α, β, and γ.

For the calculations shown in Fig. 4(a) of the main text, we set E = 0 for simplicity, as the small electric field used in the experiment, E = 18 V/cm, has a negligible effect on the state-decomposition, and we numerically calculate α, β, and γ as a function of the magnetic field.

S3. SELECTION RULES FOR NUCLEAR SPIN CONSERVATION IN ULTRACOLD

REACTIONS

Selection rules in chemical reactions arise when the S-matrix is is diagonalizable in a basis made up of the eigenstates of some conserved quantity, also known as a collision constant [START_REF] Quack | [END_REF].

Due to their weak couplings to other degrees of freedom, nuclear spins of molecules may be conserved in chemical reactions [22] and so their eigenstates may be used to derive one such selection rule. For ultracold reactions of bi-alkali molecules of the form, AB+AB → A 2 +B 2 , the nuclear spin states of the heteronuclear reactants can be described in the uncoupled basis, {|i A1 , i B1 , m A1 , m B1 ⊗|i A2 , i B2 , m A2 , m B2 }, while those of the homonuclear products are more naturally expressed in the coupled basis, {|I A 2 , M A 2 ; i A1 , i A2 ⊗ |I B 2 , M B 2 ; i B1 , i B2 }, in which the states obey the exchange symmetry of identical nuclei. Here, A1 (A2) and B1 (B2) denote atomic nuclei from the 1st (2nd) AB molecules in the reaction. The total spin is defined as I A 2 (or B 2 ) = i A1(or B1) + i A2(or B2) and its projection is

The coupled basis state |I, M ; i 1 , i 2 is symmetric under interchange of particles 1 and 2 if i 1 + i 2 -I is even, and antisymmetric if i 1 + i 2 -I is odd.

To obtain the probabilities for scattering into different channels, or product states, we can transform the reactant nuclear spin state from the uncoupled basis into the coupled basis.

The coefficients of the reactant state in the coupled basis then provide the desired information regarding the branching into the different reaction channels. This basis transformation can be performed using the general formula for the addition of angular momentum,

where M = m 1 +m 2 and I, M ; i 1 , i 2 |i 1 , i 2 , m 1 , m 2 are the Clebsch-Gordan (CG) coefficients.

Take the reactant state

for example. Assuming that nuclear spins are spectators during the reaction, we can write the reactant nuclear spin state, are antisymmetric under the exchange of identical nuclei, while all others are symmetric.

Therefore the total probability for scattering into an antisymmetric nuclear spin state of the products is given by, P A = |αβ| 2 + |αγ| 2 + |βγ| 2 , whereas that for scattering into a symmetric state is P S = 1 -P A . Furthermore, based on the exchange symmetry of identical nuclei, the populations of even (odd) rotational states of the products are thus proportional to P S (P A ) when the nuclei are bosonic, and are proportional to P A (P S ) when the nuclei are fermionic.

While this result was derived for a specific input state Eq. (S7), the methodology does not depend on the particular form of this state, so that the formulae for P A and P S can be generalized to account for any initial states of the reactants. Additionally, as far as the summed probabilities are concerned, it is not strictly necessary to consider the CG coefficients of the dressed product states in the presence of a magnetic field. Instead, it is sufficient, for the purposes of this work, to consider only the CG coefficients of the bare states in the absence of a field. However, for a state-to-state measurement which resolves individual nuclear spin states, it is, strictly speaking, necessary to consider the CG coefficients of the dressed states. This will be reported in another work.


« The Cultural Caravan for Peace: a controversial peacekeeping project on the roads of Malian political crisis »

Introduction

In 2012, the political situation in Mali goes to large disruptions: development and economical issues entail vehement political opposition in the South. A military coup occurs on March 22 nd , and President Amadou Toumani Touré resigns after ten years of presidential term.

In the North, reminiscences from the last great Tuareg rebellion in the 90's merge with the recent phenomenon of Islamic extremism spread over the Sahara; rebel armed groups -some of them being jihadist -take advantage of the confusing situation and overrun the area, disputing control for the main cities (Timbuktu, Gao, Menaka, Kidal). The 6th April 2012, the MNLA declares the independence of "Azawad", a territory covering the entire northern region of Mali. Ansar Eddine, one of the main jihadist groups, imposes Islamic law "Chari'a" on the whole Azawadian territory and applies violent treatment to anyone who would break it. Facing this deleterious security context, more than 200 000 northern inhabitants leave their home to flee oppression. They conglomerate into refugee camps temporarily built up at the borders with Mauritania, Algeria and Burkina Faso, expecting solutions in order to lastingly set up in the South, mostly in Bamako. In January 2013, while Ansar Eddine attempt to take hold of Southern parts of the country, the French army stop them under Serval military action required by the interim president. The United Nations involve into the conflict in order to pacify by sending international military forces (MINUSMA) and help with resetting the Malian state. New elections are organized height months later, and a peace-building process is implemented under a multilateral dialogue1 .

Whereas the immediate key factors of the crisis appeared to be linked with economical and political troubles, the global composition of the national population results to a large cultural diversity to manage on a unique territory: 60 different ethnic groups, and more than 35 languages 2 . While institutional rhetoric consists in promoting this cultural diversity as a national wealth, the daily reality of coexistence implies much mutual understanding of one another's way of life, as well as dedicated policies to rule proper governance.

Within such a politico-cultural context, a variety of initiatives aim at enhancing intercultural dialogue: this is the case of art and culture organisations offering cultural events in order to increase a common feeling of tolerance between people. Are these cultural promoters and mediators able to sooth tensions in a situation of armed conflict and political violence? This presentation deals with the Cultural Caravan for Peace, a particular peacekeeping project of these, which was created as a cultural reply to the war. We will examine its capacity to react to the conflict situation in order to disseminate a reconciliation message.

I.

Origins and creation of the Cultural Caravan for Peace A. Before the Caravan in the South, a festival in the North

The Cultural Caravan for Peace appeared as a consequence of the crisis, which broke out in the North in 2012. Indeed, the organization committee, especially the director of the Caravan, were among the northern refugees who went in exile to the South. Manny Ansar and his team used to organize a festival entitled the Festival in the Desert for years (2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012), first in Kidal area and then in Timbuktu region. This festival, initiated by French and Tuareg musicians and cultural promoters following the disarmament and peace-making process in the late 90's, was internationally known and grew year by year until it became one of the largest African cultural gathering, taking place in the middle of Saharan dunes. More than ten thousands visitors attended the event every year. Famous artists such as Bono (U2), Robert Plant (Led Zeppelin), or Manu Chao, shared the stage with Malian well known artists like Tinariwen, Salif Keita, Ali Farka Toure and more. The festival turned to a mean to support development in the North, to promote Tuareg culture and values at a national and international scale, and to favour intercultural dialogue between people. Whereas international tourists came to discover Tuareg culture and make an original experience in the desert, Manny Ansar had to ensure safety for every participant. However, the larger the festival became, the more oppressive the threats were to him3 . In 2012, the last edition of the festival was held a few weeks before the political outbreak in a very tensed atmosphere, punctuated by security alerts4 . In 2013, when the Islamic armed groups took control of the area, music and festivities were totally prohibited. The festival could no longer exist in its sedentary shape in the North.

B. Using nomadism to promote peace on Malian roads

When Manny Ansar reaches the South with his relatives, he finds new support from other cultural promoters and institutional actors. Unable to return home, they decide to create a new artistic event, reflecting a political statement in favour of peace and unity in Mali: the Cultural Caravan for Peace.

As a way to keep the vocation of giving visibility to Northern cultures, but also to address the need to wisely promote a peaceful message to Southern communities, a nomadic style is applied to the event. Its itinerary drives every year, from 2014 to 2019, toward the North of the country, as a symbolic path and wish. Stops are planned in strategic cities, whether they can be the location of partner event (Festival on the Niger in Segu) or strategically situated (Bamako, Mopti, Niafunké) on a political approach.

II.

Activism for Peace through Music A. Bringing joy and hope to disparate people

The personal experience of the organization committee taught them about the emergency of carrying about thousands of people still stuck in the refugee camps. These people have lost everything, even sometimes relatives who count among injuries of a strict Chari'a implementation in the North. In 2014, the Caravan Cultural for Peace starts its annual trip by visiting these camps in Burkina Faso. Artists and promoters deliver a vibrant and supportive message on their common belief in a safe and pacified future, reminding that actions are made everyday to solve the political issue and help refugees to get better living-conditions5 .

In the same way, the Caravan drives until remote villages6 , where development and economical issues are everyday struggles, where cultural melting pot barely exists and ethnic confusion is quickly made. Further than advocating for Malian wealthy cultural diversity, the speeches explain the importance of education for children, solidarity and partnership between individuals and the virtue of leaving together.

B. Intercultural dialogue as the key for peace-making

Speeches are illustrated by performances given by famous artists and young talents, all representing the variety of ethnic groups of the country. Thanks to percussive shows, from traditional to modern styles of Music, they sensitize the public, always counting thousands of people, to the political message of the Caravan.

On the top of this eclectic program, the Caravan organization committee initiated the creation of a band composed of as much musicians as the number of regions in the country, each coming from one of these regions. The band was called "Malikanw", meaning "the Voices of Mali", standing for "le Mali Un et Indivisible" that is to say "A unified and indivisible Mali" 7 . They sing peace in different languages 8 , in order to be understood by any Malian citizen in the public. Sharing a common idea of fighting against the war, they use "their mouth as a bomb" 9 to discredit armed attacks and violence perpetuated on civilians. Nevertheless, in this preconceived musical project aiming at staging cultural diversity, tensions between musicians are frequent, and the artists explain backstage that they need to overcome their difference to keep on playing together10 . Musically speaking, they admit that a Malinké won't be able to play the proper guitar rhythm of Tuareg blues, and a Bobo will never rightly sing chores as expected in Wassulu chants; Malikanw and the Caravan can therefore be perceived as relevant representations of the Malian cultural context.

III.

Doubts and scepticism on the evolution of the Caravan A. A politicised project Created in a highly tensed political situation, the Caravan Cultural for Peace initially took part to the context with a clear peaceful vocation. Guided by the need to survive, the promoters of the event reached public institutions to apply for support, communicating on their claim for peace and reconciliation. They converted international funding previously allocated to the Festival in the Desert toward the Cultural Caravan for Peace, maintaining their partnerships with organization such as: Embassy of France to Mali11 , Embassy of the Netherlands to Mali, Embassy of Norway to Mali, Doen Foundation, GIZ Germany etc.

Moreover fell the Caravan within the framework of ministerial policies to promote intercultural dialogue and reconciliation between North and South, rebels and government, nomadic and sedentary populations. Under the joint distinguished patronage of the Minister of Culture, Craftwork and Tourism, and the young Minster for Reconciliation, the Caravan was given an official recognition. President Ibrahim Boubacar Keita and his spouse themselves were loyal supports and attended shows in Bamako. In 2015, few weeks after the Caravan drove from Bamako to Mopti, the Agreement of Algiers, defining the terms for new governance in Mali taking into account particularities of Azawad area, was signed between the opposite parties.

Speech of Ms Ndiaye Ramatoulaye Diallo, Minster of Culture, craftwork and tourism of Mali, Bamako, Cultural Caravan for Peace 2015 : « Avant de saluer l'initiative, je l'encourage à grandir dans un contexte autre. Je l'encourage à dire pour évoquer ce qui devrait être notre préoccupation première : le développement. A travers le dynamisme et la créativité de ses initiateurs, cette Caravane de la Paix, tout en symbolisant la volonté de tout un peuple, s'impose comme un véritable espace de dialogue, d'échanges et d'expression artistique et culturelle, propice à la vulgarisation du message de la paix et de la réconciliation.

Mesdames et Messieurs, je note avec satisfaction et espoir la présence parmi nous de Madame la Première Dame du Mali. Cet accompagnement est loin d'être le fruit d'un hasard heureux. Il s'agit là de la conviction suivant laquelle la Culture du Mali doit être à pied d'oeuvre pour réconcilier les fils et les filles de ce pays. » This political enrolment of the event entailed much criticism from previous participants of the Festival in the desert, and a part of northern citizens, especially in Kidal area. Some famous artists choose to boycott the Caravan and denied the invitation to perform. Indeed, the Caravan being associated to national institutions appeared as mobilized by the government to reassert its authority on the northern region. The promoters and some of the artists performing during the Caravan, despite their Tuareg origin, were considered as political supporters, betraying their relatives who fought for the autonomy of Azawad.

B. Access denied: the Caravan stuck in the South

The closed partnerships with the government and the international community advocating for the unity of Mali made the Caravan an enemy of the Tuareg rebellion. Despite the good will of the organizers to favour dialogue and negotiation, their political statement warded them off from northern populations who still suffer from the lack of development on one hand, and violence from both jihadists still circulating in the area and the Malian army on the other hand. Year after year, the Caravan strived to drive upper and upper in direction to the North. In 2016, as the itinerary should finish in Gao, the convoy was stopped at the doors of Timbuktu and demanded to drive back to the South12 . From then, the project has been consequently reduced in its official version: a few shows are still organized in the framework of other festivals in Mali (Segu) and abroad (Morocco, Belgium). Cut off from its roots and goals, the Caravan can no longer undertake its role.

Conclusion

Promoters of the Cultural Caravan for Peace live a very intricate situation: whereas they left Northern Mali to go to exile in the South and created a project inspired from northern traditions, they could never go back home to spread the peaceful message where people mostly need to hear it. While the Caravan has wisely been publicized in the years of the Algiers Agreement process, it nowadays surreptitiously produces shows on the occasion of other festivals, most of them abroad. As security has not been completely restored in Mali and the Sahelo-Saharan area, Peace remains hard to implement and its main actors look breathless.

We have observed that war situations can create hopeful initiatives to help with pacifying a territory when discordance and violence disrupt a country global governance and the national communities living-conditions. Founding on the cultural dimension of a conflict, civilians offer cultural projects as peace-making means. However, according to the evolution of the political context, making peace doesn't mean solving the conflict with satisfying solutions for the opposite parties. With or without their will, cultural promoters get related to funding institutions and partners, playing themselves a political role in the conflict.

More than being neutral mediators, they therefore become, in a way, stakeholders within the conflict. Then have they to overcome impediments and threats, to keep their project and faith alive.

For more details on Malian crisis, see T. PERRET, Mali, une crise au Sahel, Karthala, Paris,

2 J. LECLERC. « Mali » in L'aménagement linguistique du monde, Québec, CEFAN, Université Laval, 15 avril 2015, [online] http://www.axl.cefan.ulaval.ca/afrique/mali.htm accessed on May 15th, 2019. Laetitia Chantrait Ph. D. candidate 3 rd year -Modern History of Africa Ecole Doctorale d'Histoire -Institut des Mondes Africains Université Paris 1 Panthéon-Sorbonne

Interview with Manny Ansar, Bamako, Mali, 2017.

M. AMICO, La Fabrique d'une « Musique Touarègue ». Un « son du désert » au prisme de la World Music, Ph. D. thesis in Anthropology under the supervision of Prof. Denis Laborde, EHESS, Paris, 2013 ; S. Daniel, « A Tombouctou, un festival avec la star Bono fait oublier Al-Qaïda », published on January 14 th , 2012, [online] https://www.ladepeche.fr/article/2012/01/14/1260788-a-tombouctou-un-festival-avec-lastar-bono-fait-oublier-al-qaida.html, accessed on January 4 th , 2017.

Speech of Manny Ansar, Cultural Caravan for Peace,Mopti, Mali, 2015.

Sélingué (2014); Koutiala, San (2015); Diré, Bourem (2016).

National motto of the country

Bambara, Songhoy, Peul, Dogon, Tamashaq etc.

Interview with Malikanw, Samba Touré, Cultural Caravan for Peace, Segu, Mali, 2015.

Interview with Malikanw, Petit Goro, Cultural Caravan for Peace, Segu, Mali, 2015.

Interview with Yves Gueymard, former cultural counsellor for the Embassy of France to Mali, France, 2019.

Interview with Khaira Arby, artist, Bamako, Mali, 2017.

Short Abstract

The Cultural Caravan for Peace was created as a peacekeeping mean, promoting political reconciliation to face the aftermath of the Malian political crisis. This proposal aims at underlining the paradoxical position of cultural actors in a conflict situation, between neutrality and involvement.

Abstract

When the Malian crisis starts in 2012 with the military coup and the occupation of the North by rebel armed groups -mostly jihadists groups, more than 200 000 northern inhabitants urgently leave their home to flee oppression and violence. Founded on former claiming for independence from Touareg populations, the conflict strengthens a feeling of division within the whole country, between main geographical and ethnical areas.

As per a reaction to this situation, a group of artistic promoters decide to launch the Cultural Caravan for Peace. This peacekeeping mean aims at circulating a federating message, promoting intercultural dialogue and political reconciliation through shows to be given in different affected places in Mali and around. Every stop on the road appears as a quiet and joyful musical meeting point between different ethnic groups composing the public for one night, despite rampant surrounding threats and hazards. The peaceful word is widely spread thanks to the artist's fame and international medias. But the Caravan becomes institutionalized year by year. A large portion of the northern communities, including rebel groups, who are still favourable to obtain the autonomy of their Saharan region, doubt over its initial political neutrality.

Until today, the Cultural Caravan for Peace has surely impacted people's minds everywhere it passed by, sensitizing about mutual understanding. However, although cultural actors would rather stay on the side of the conflict, they finally get involved in actions reflecting their political opinion. This proposal aims at underlining the paradoxical position of cultural actors in a conflict situation.