D Bathellier

L Raymond

Thibaut Jonckheere

Jérôme Rech

A Zazunov

T Martin

Finite frequency noise in a normal metal-topological superconductor junction

come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

The search for Majorana fermions 1 -particles that are their own antiparticles -has been an active field of study in high energy physics, with no conclusive/definite evidence of their finding. Since the last decades, the Majorana paradigm has now entered condensed matter physics thanks to the pioneering work of Kitaev. 2 In this context, rather than being elementary particles, Majorana fermions emerge from the collective behavior of a many electron system. This Kitaev toy model consists in a tight-binding chain of electrons living on lattice sites, which includes hopping and superconducting (p-wave) pairing between neighboring sites. Majorana fermions exist at the boundaries of this one dimensional chain provided that certain conditions on the hopping parameter, the gap parameter, and the chemical potential are met. In this so-called topological phase, the ground state of the system is degenerate, opening the possibility to realize a quantum bit which could be in principle immune from decoherence effects, with applications to quantum information schemes.

In quantum nano-physics, a huge effort has been devoted to finding a realistic experimental equivalent of the Kitaev model. One possibility is to realize a topological superconductor (TS) nanowire by inducing s-wave superconductivity in a semiconducting nanowire subject to both Rashba spin-orbit coupling and a Zeeman magnetic field. [3][4][5][6][7][8] Other proposals where a chain of magnetic atoms are deposited on a superconducting substrate have also been studied. 9 Many theoretical proposals for the detection of Majorana fermions relying on quantum transport setups have been published over the past few years, [10][11][12][13][14][15] but despite the significant experimental progress, an unequivocal signature 16,17 of these Majorana bound states (MBS) is still lacking. While some elements point in the right direction (the detection of the zero bias anomaly in the differential conduc-

-Δ Δ E ρ(E) L R C N T S V Figure 1
. Top: schematic view of a junction between a normal metal (yellow dots) and a topological superconductor (redgreen dots, with a red Majorana bound state at its end); the normal metal is biased at voltage V . Bottom left: topological superconductor density of states, with a central peak at zero energy due to the Majorana bound state, and square root behavior outside the gap. Bottom right: the principle of the noise detection, using the inductive coupling to a RLC circuit. 20 tance, and quite recently a claim that the quantized zero bias conductance has been observed 18 ) it is clear that the definite discovery of Majorana fermions will require more experimental tests in order for this finding to be firmly established. This is especially important in order to exploit the properties of these objects in the future, such as the generation of a Majorana qbit, and the braiding of Majorana fermions, with applications to quantum computing.

In this work, the physics of a normal metal -topological superconductor (NTS) junction (see Fig. 1) is revisited. While the non-equilibrium electronic current and the zero-frequency noise are perfectly understood by now, 19 the finite frequency noise characteristic of a NTS junction has so far not been considered. Finite frequency noise has been studied extensively in more conventional materials in the context of quantum mesoscopic physics and nano-physics, both theoretically and experimentally. On the theoretical side, it has been argued [20][21][22] that since the two current operators entering the noise correlator are evaluated at different times, one can introduce two quantities -the emission and the absorption noise. A noise detector should in principle be described on the same quantum footing as the nano-device under study. In full generality, the Fourier transform of the real time noise correlators can be a complex function of the frequency. The measured signal by the quantum detector, which is a real number, should thus be a combination of such emission and absorption quantities. This has been experimentally verified in pioneering finite frequency measurement experiments performed in Deflt, 23 in Orsay [24][25][26] and in Saclay [27][28][29] in different contexts. In a previous work, one of us 30,31 computed the finite frequency noise of a normal metal -BCS superconductor junction, and found cusps of singularities at specific frequencies corresponding to different transport processes. However, these results are not expected to survive in the case of the NTS junction, as the density of states (DOS) of the topological superconductor differ substantially from its conventional BCS counterpart. The BCS density of states is zero inside the gap, and bears inverse square root divergences at the gap edges. On the opposite, the DOS of a TS wire has a square root behavior outside the gap and a so-called Majorana peak at zero energy (see Fig. 1).

Using a continuous model describing a semi-infinite Kitaev chain, we derive the finite frequency noise to all orders in the tunnel coupling parameter of the NTS junction (recovering in the process known results for the current and zero-frequency noise 19 ). Within our framework, the current and noise can be expressed in terms of dressed Keldysh Green's functions (using Wick's theorem for the noise as the Hamiltonian is quadratic). Solving Dyson's equation exactly, one reaches analytical expressions for these two quantities in terms of the known bare boundary Green's functions of the two leads. The finite frequency emission and absorption noises are then computed for voltages above and below the superconducting gap, showing plateaus -with a specific frequency width -where the latter quantities are essentially constant. We interpret such features as emission and absorption processes which couple the constant density of states of a normal metal with the sharp peak in the TS density of states associated with the Majorana bound states. For completeness, we further use a generic model for the quantum detector -a LC circuit which can absorb or emit photons from or to the nano-device 20,22,32 -and compute the measurable noise at the resonant frequency of the LC detector.

In order to illustrate the added value of using finite frequency noise over other quantities relying on DOS, like the differential conductance, we have also computed the transport properties of a non-topological N-dot-S system, where S is a standard BCS superconductor. Indeed this system bears a zero-energy Andreev bound state, with a sharp peak in the DOS, which is susceptible to yield similar differential conductance and finite frequency noise profiles. With the right parameters, this system presents a conductance peak similar to the one of a Majorana bound state, which shows that the conductance alone is not enough to characterize a MBS. However, the emission noise shows a distinctive dip near zero-frequency in this system, which is absent for the NTS junction. We thus believe that the finite frequency noise characteristics of an NTS junction can bring unique, additional information about Majorana fermions in experiments. This constitutes the main motivation of this study.

The paper is organized as follows: Sec. II introduces the microscopic model Hamiltonian and defines the Keldysh Green's functions. The Dyson's equations which are relevant for the current and noise calculations are specified in Sec. III. The noise is computed in Sec. IV (real time correlator, emission and absorption noise,...) and detailed results are discussed in Sec. V. Sec. VI shows the calculations and the results for the emission noise of a N-dot-S system, with a BCS superconductor. We conclude this work with Sec. VII. In the whole following text, we fix the value of to 1.

II. MODEL

A semi-infinite topological superconductor wire is a single-channel p-wave superconductor with spinmomentum locking, described from the continuous version of the Kitaev chain 2 by the helical superconductor Hamiltonian

H j = +∞ 0 dx ψ † j (x)(-iv F ∂ x σ 3 + ∆σ 2 )ψ j (x) , (1)
where v F is the Fermi velocity, ∆ is the superconducting gap, and σ 1,2,3 are Pauli matrices in Nambu space. Here, we refer to j = 0 for the TS nanowire and j = 1 for the (spinless) normal metal lead, whose Hamiltonian has the same form, with the gap parameter set to zero. These lead Hamiltonians are conveniently written in terms of the fermionic Nambu spinor ψ j defined as

ψ j =   c Rj c † Lj   , (2)
which combines left-and right moving fermion operators c R/L,j . The Hamiltonian describing the tunnel coupling between the two leads has the form (in the absence of applied bias)

H T = λc † 0 c 1 + H.c. , (3)
where c 0 and c 1 are boundary fermions built simply from the sum of the left and right fermionic fields entering the Nambu spinor, evaluated at the location x = 0 of the NTS junction, so that c j = c Rj (0) + c Lj (0). It is also convenient to write this tunnel Hamiltonian in the Nambu representation. The formalism can be generalized to an arbitrary number M of tunnel-coupled conductors as in Ref. 33. For this, we define a tunneling matrix W in lead space, whose diagonal elements vanish:

W jj = λσ 3 (1 -δ jj ).
Then, in Nambu representation, the tunneling Hamiltonian takes the form

H T = 1 2 M jj Ψ † j W jj Ψ j , Ψ j =   c j c † j   . (4)
The operator accounting for the current flowing through lead j, is derived from the tunneling Hamiltonian (4) as

I j = ie j =j Ψ † j σ 3 W jj Ψ j . (5)
An essential tool for this study is the Keldysh Green's function for the boundary fermions. It is defined as

Gηη jk (t, t ) = -i T K c j (t η )c † k (t η )
, where η = ± denotes the Keldysh time index which specifies the position of times on the Keldysh contour. It is represented by the (4 by 4) matrix in Keldysh-Nambu space:

Gjj = G ++ jj G +- jj G -+ jj G -- jj = L 0 G a jj G r jj G K jj L-1 , (6)
where G a , G r , G K are the advanced, retarded and Keldysh components in Nambu space, and the unitary transform L = (1/ √ 2)(τ 0 + iτ 2 ) ⊗ σ 0 (τ j are Pauli matrices in Keldysh space).

We wish to express the average current in terms of such Green's functions. For this we write explicitly the expression of the off-diagonal Keldysh Green's functions:

G -+ jj (t, t ) = -i c j (t)c † j (t ) c j (t)c j (t ) c † j (t)c † j (t ) c † j (t)c j (t ) (7)
G +- jj (t, t ) = +i c † j (t )c j (t) c j (t )c j (t) c † j (t )c † j (t) c j (t )c † j (t) (8)
The average current can then be recast 19 in the form of a Nambu trace containing the Keldysh Green's function G K jj (t, t ) = G +- jj (t, t ) + G -+ jj (t, t ):

I j = e 2 j =j Tr N σ 3 W jj G K j j (t, t) , (9)
which reads, in Fourier space, for the two-lead geometry chosen here

I(V ) = eλ 2 +∞ -∞ dω 2π Tr N G K 01 (ω) , (10)
where we used that, in this case, the tunneling matrix reduces to W 01 = W 10 = λσ 3 .

III. DYSON'S EQUATIONS AND AVERAGE CURRENT

The Dyson's equation is readily expressed in Keldysh-Nambu-lead space as

Ǧ = ǧ-1 -W -1 , (11)
where we have

Ǧ = G00 G01 G10 G11 , W = W00 W01 W10 W11 , (12)
and the tunneling self-energy Wjj is a diagonal matrix of the form Wjj = diag (W jj , -W jj ). Using Eq. ( 6), this can be further expressed into a set of equations for the advanced, retarded and Keldysh Green's functions as Ĝr/a = ĝr/a + ĝr/a Ŵ Ĝr/a (13) ĜK = 1 + Ĝr Ŵ ĝK 1 + Ŵ Ĝa (14) where we used that the Keldysh component associated with the tunneling self-energy vanishes and introduced the notation • for matrices in Nambu-lead space.

In order to compute the current, we thus need to know the full expression of the dressed Green's function G K 01 . The Dyson's equation allows us to express this Keldysh component in terms of the advanced and retarded dressed Green's functions as well as the bare Keldysh Green's function and the tunneling self-energy. One has

G K 01 = (σ 0 + G r 01 W 10 ) g K 00 W 01 G a 11 + G r 00 W 01 g K 11 (σ 0 + W 10 G a 01 ) . (15)
A. Bare Green's functions

As a first step, we need to specify g K 00 and g K 11 . The procedure to obtain the boundary Green's function for the uncoupled semi-infinite TS wire has been described carefully in Ref. 19. Following this derivation, the various components of the bare Green's functions for the topological superconductor then read

g K 00 (ω) = [1 -2n F (ω)] [g r 00 (ω) -g a 00 (ω)] (16) g r/a 00 (ω) = ∆ 2 -(ω ± i0 + ) 2 σ 0 + ∆σ 1 ω ± i0 + , (17)
where n F (ω) is the Fermi function and 0 + an infinitesimal introduced to properly define the square root in the complex plane. The Keldysh, retarded and advanced bare Green's functions for the metallic lead are readily obtained by setting ∆ = 0 in the results above, so that

g K 11 (ω) = -2iF 1 (18) g r/a 11 (ω) = ∓iσ 0 . (19)
There we have introduced the matrix F 1 as

F 1 = 1 -2n F (ω -eV ) 0 0 1 -2n F (ω + eV ) . (20)
Note that the top and bottom diagonal elements respectively have an electron-like and a hole-like character, thus leading to opposite signs for the voltage.

Introducing the Heaviside distribution Θ(|ω| -∆), the boundary Keldysh Green's function on the TS side becomes

g K 00 (ω) = 2i tanh βω 2 × π∆(σ 0 + σ 1 )δ(ω) + √ ω 2 -∆ 2 |ω| Θ(|ω| -∆)σ 0 . ( 21
)
where β is the inverse temperature.

B. Average current

In order to write the average current, we then need the expression for the dressed Green's functions G

G r/a

10 (ω) = ∓iλσ 3 G r/a 00 (ω) , (23)
G r/a 11 (ω) = ∓iσ 0 -λ 2 σ 3 G r/a 00 (ω)σ 3 . ( 24
)
It follows that the Keldysh component of the dressed Green's function which enters the current average can be written as

G K 01 = iλ σ 0 -iλ 2 G r 00 g K 00 σ 0 + iλ 2 G a 00 σ 3 -2iλG r 00 F 1 σ 0 + iλ 2 G a 00 σ 3 . (25)
Where G r/a 00 is also obtained from the Dyson's equation, and reads

G r/a 00 = -σ 3 g r/a 00 σ 3 ± iλ 2 σ 0 -1 . ( 26
)
One readily sees that G r/a 00 is a linear combination of σ 0 and σ 1 Pauli matrices

G r/a 00 (ω) = α r/a (ω)σ 0 + γ r/a (ω)σ 1 , (27)
where, denoting

ω ± = (ω ± i0 + ), α r/a (ω) = ∆ 2 -ω 2 ± ∓ iλ 2 ω ± ω ± ∆ 2 - ∆ 2 -ω 2 ± ∓ iλ 2 ω ± 2 , (28)
γ r/a (ω) = ∆ω ± ∆ 2 - ∆ 2 -ω 2 ± ∓ iλ 2 ω ± 2 . (29)
Note that α r = (α a ) * and γ r = (γ a ) * . Keeping in mind that g K 00 is also a linear combination of σ 0 and σ 1 , one can significantly simplify the Nambu trace needed for the current

Tr N G K 01 (ω) = -2iλTr N G r 00 F 1 σ 0 + iλ 2 G a 00 σ 3 . (30)
Inserting all these expressions in the trace formula for the current Eq. ( 10), the result is, introducing

n F (ω ± eV ) ≡ n (±) F and δn F = n (+) F -n (-)
F :

I(V ) = -2ieλ 2 +∞ -∞ dω 2π α r + iλ 2 |α r | 2 -|γ r | 2 δn F .
(31) This is identified as a Landauer formula

I = e 2π +∞ -∞ dω R A (ω) n (-) F -n (+) F . ( 32
)
with the (real valued) Andreev reflection probability

R A (ω) = -2λ 2 Im (α r ) + λ 2 |α r | 2 -|γ r | 2 , (33)
whose expression matches exactly with the one of the spectral current density J(ω) obtained in Eq. ( 35) of Ref. 19.

IV. FINITE FREQUENCY NOISE

A. Real-time noise correlator

The real time noise correlator in a multi-lead geometry is defined as:

S jl (t, t ) = I j (t)I l (t ) -I j (t) I l (t ) , (34)
where I j (t) is defined by Eq. ( 5). As the Hamiltonian is quadratic in the fermionic degrees of freedom, we can use Wick's theorem to express the four fermion operator averages in terms of products of off-diagonal Keldysh Green's functions, again in the form of a Nambu trace:

S jl (t, t ) = e 2 Tr N σ 3 Ŵ Ĝ-+ (t, t ) Ŵ jl σ 3 G +- lj (t , t) -e 2 Tr N σ 3 Ŵ Ĝ-+ (t, t ) jl σ 3 Ŵ Ĝ+-(t , t) lj . ( 35
)
where we recall that • corresponds to matrices in Nambulead space.

In our two-terminal geometry, this simplifies as

S 11 (t, t ) = λ 2 e 2 Tr N G -+ 00 (t, t )G +- 11 (t , t) -G -+ 01 (t, t )G +- 01 (t , t) . (36)
These Green's functions are related to the retarded, advanced and Keldysh components according to Ĝ+-= 1 2 Ĝa -Ĝr + ĜK ,

Ĝ-+ = 1 2 Ĝr -Ĝa + ĜK . (37)
Writing explicitly the Keldysh component of the Dyson's equation, Eq.( 14), for all combination of leads, one has

G K 00 = σ 0 -iλ 2 G r 00 g K 00 σ 0 + iλ 2 G a 00 -2iλ 2 G r 00 F 1 G a 00 ( 39
)
G K 11 = (σ 3 -iλ 2 σ 3 G r 00 )(λ 2 g K 00 -2iF 1 )(σ 3 + iλ 2 G a 00 σ 3 ) (40)
The four Green's functions G -+ 00 , G -+ 01 , G +- 01 , G +- 11 which enter the real-time noise correlator are ultimately obtained from the above Keldysh components

G -+ 00 = 1 2 G r 00 -G a 00 -2iλ 2 G r 00 F 1 G a 00 + H , (41)
G +- 11 =i(σ 0 -F 1 ) + λ 2 2 σ 3 (G r 00 -G a 00 ) σ 3 -λ 2 σ 3 G r 00 σ 3 F 1 + λ 2 F 1 σ 3 G a 00 σ 3 -iλ 4 σ 3 G r 00 F 1 G a 00 σ 3 + 1 2 λ 2 σ 3 Hσ 3 , (42)
G -+ 01 = -i λ 2 (G r 00 + G a 00 ) σ 3 -iλG r 00 σ 3 F 1 + λ 3 G r 00 F 1 G a 00 σ 3 + i λ 2 Hσ 3 , (43)
G +- 01 =i λ 2 (G r 00 + G a 00 ) σ 3 -iλG r 00 σ 3 F 1 + λ 3 G r 00 F 1 G a 00 σ 3 + i λ 2 Hσ 3 , (44)
where we introduced the new matrix

H(ω) = σ 0 -iλ 2 G r 00 g K 00 σ 0 + iλ 2 G a 00 . (45)
Substituting these back into the expression for the realtime current correlator allows one to derive an exact, albeit cumbersome, analytic expression for S 11 .

B. Emission, absorption and measurable noise

In full generality, one may consider that the Green's functions depend on two times t and t . In the specific setup under consideration, we are in a stationary situation as there is only one superconductor in the system. It follows that the time dependence reduces to a single variable, more precisely the time difference τ = t -t .

We thus consider the two distinct correlators:

S + (τ ) = S 11 (0, τ ) = λ 2 e 2 Tr N G -+ 00 (-τ )G +- 11 (τ ) -G -+ 01 (-τ )G +- 01 (τ ) , (46)
S -(τ ) = S 11 (τ, 0) = λ 2 e 2 Tr N G -+ 00 (τ )G +- 11 (-τ ) -G -+ 01 (τ )G +- 01 (-τ ) , (47)
Their Fourier transforms define the emission and absorption noise as

S + (Ω) = +∞ -∞ dt δI(0)δI(t) e iΩt , (48)
S -(Ω) = +∞ -∞ dt δI(t)δI(0) e iΩt . ( 49
)
which are conveniently rewritten using the Fouriertransformed Green's functions as

S + (Ω) = λ 2 e 2 +∞ -∞ dω 2π Tr N G -+ 00 (ω)G +- 11 (ω + Ω) -G -+ 01 (ω)G +- 01 (ω + Ω) (50) S -(Ω) = λ 2 e 2 +∞ -∞ dω 2π Tr N G -+ 00 (ω)G +- 11 (ω -Ω) -G -+ 01 (ω)G +- 01 (ω -Ω) . (51)
It is clear from these expressions that emission and absorption noises are trivially related when flipping the sign of the probing frequency, namely S + (Ω) = S -(-Ω). It follows that in order to describe the whole range of physical parameters, we can safely focus on positive frequencies for both noise spectra.

In order to connect with potential experimental works looking to investigate the finite-frequency noise of the NTS junction, we also compute the noise expected to be measured from a generic quantum detector consisting of a resonant LC circuit, inductively coupled to the NTS junction (see Fig. 1). This measured noise is the result of repeated measurements of the charge at the capacitor plates. For the present setup, we use an expression formulated in Refs. 20 and 32 to predict the result of such a measurement. This so-called measurable noise is given by

S meas (Ω) = K{S + (Ω) + N (Ω)[S + (Ω) -S -(Ω)]} , (52)
where

N (Ω) = [exp( Ω/k B T LC ) -1] -1
is the Bose-Einstein distribution associated with the oscillator modes, and K is the effective coupling constant of the quantum wire with the resonator whose expression reads

K = α 2L 2 1 2η . ( 53
)
L and α respectively stand for the inductance of the resonant circuit and the inductive coupling, while η is the width of the resonance. This last parameter depends on the dissipative environment surrounding the LC circuit, 32 and is typically small but its inverse is kept finite. Note that in full generality, this measurable noise is a combination of emission and absorption noise, weighted by the Bose-Einstein distribution. The temperature of the noise detector T LC can in general be different from that of the electrons flowing in the NTS junction.

V. RESULTS

In this section, we start by briefly recalling the known results for the conductance and the zero frequency noise, as both these quantities serve as a useful benchmark for our approach. We then move on to the determination of the finite frequency emission, absorption and measurable noises at low temperature.

The NTS junction is characterized by its transparency τ (0 ≤ τ ≤ 1), which is defined from the tunneling amplitude λ as τ = 4λ 2 /(1 + λ 2 ) 2 . We use units such that k B = = 1. We adopt the convention that the TS wire is grounded, the voltage drive being applied to the normal metallic lead.

A. Differential conductance

We hereby recover important results which were already obtained in a previous study. 19 At zero temperature, the differential conductance is directly given by the reflection probability G = 2e 2 h R A (eV ) [see Eq. ( 32)]. At subgap voltages, this takes a very simple form since in this case

R A (eV ) = 1 1 + eV Γ 2 , (54)
leading to the predicted Majorana zero-bias peak with quantized height 2e 2 /h and width Γ = τ ∆/2 √ 1 -τ due to resonant Andreev reflexion. As one approaches perfect transmission τ → 1, the strong NTS hybridization turns this peak into a plateau at G = 2e 2 /h extending over the whole subgap regime, |eV | < ∆ (although the Majorana state is ill-defined due to this very same hybridization). Moreover, setting ∆ = 0, one recovers the behavior of a spin polarized N-N junction with a Landauer conduc-

tance G N N = τ e 2 h
. Adding a finite temperature to the system does not substantially change the results and only tends to smooth out the curves, as expected.

B. Zero-frequency noise

We now turn to the shot noise S 11 = S ± (Ω = 0), and more specifically the Fano factor, defined as the ratio of the zero-frequency noise to the average current, F = S 11 /eI 1 .

In the subgap regime, as predicted in Ref. 19, this Fano factor vanishes at perfect transparency (τ = 1), and increases when the transparency decreases following

F = 1 - 1 arctan eV Γ eV Γ 1 + eV Γ 2 . ( 55
)
Above the gap, even at τ = 1, the Fano factor F is finite due to the concomitant presence of both Andreev reflexion and quasiparticle processes.

For a normal-normal junction, i.e. in the limit ∆ = 0, the zero-frequency noise at zero temperature shows, as expected, 34,[START_REF] Martin | Noise in Mesoscopic Physics, Nanophysics : coherence and transport : école d'été de physique des Houches[END_REF] a linear behavior with respect to the bias voltage and reduces to S 11 = (e 2 /h)τ (1 -τ )eV .

C. Finite-frequency noise

The finite-frequency noise has so far been absent in the investigations of the transport characteristics of topological superconductors. The results for the emission noise S + (Ω) and the absorption noise S -(Ω), obtained from a numerical evaluation of Eqs. ( 50) and (51), are shown in this section. Only their real parts are discussed here, as their imaginary parts yield vanishing contributions within numerical accuracy.

The noise at finite frequency Ω is created by fluctuations of current processes which are accompanied by the emission or absorption of a photon at the frequency Ω. Understanding the basic process which contributes to the current with emission or absorption of a photon, while conserving the total energy, can give us a qualitative understanding of the finite-frequency noise features.

As a warm-up, let us first recall what happens in the most simple situation of a junction between two normal metallic electrodes (corresponding to taking the limit ∆ → 0). There, electronic transport occurs if occupied electron states from one lead have enough energy to reach the empty states of the other. This results in a decreasing linear behavior of the emission noise S + (Ω) for frequencies Ω ∈ [0, eV ] and a vanishing emission noise for higher frequencies due to Pauli blocking. [START_REF] Martin | Noise in Mesoscopic Physics, Nanophysics : coherence and transport : école d'été de physique des Houches[END_REF] In the meantime, the absorption noise S -(Ω) increases linearly with Ω, since the bigger the frequency, the more electrons can be transfered between leads.

When we replace the normal lead by a topological superconducting wire, the noise spectrum changes drastically. These changes can be understood qualitatively by taking into account the density of states of the topological superconductor, which is gapped for energies |E| < ∆, except for a single narrow peak at zero energy due to the Majorana bound state (see Fig. 1).

Qualitative picture

The qualitative picture showing the basic processes contributing to the finite-frequency noise are shown in Fig. 2. The two processes of the top row (a and b) describe emission noise, while the four other processes describe absorption noise. Note that we can choose eV > 0 without loss of generality, thanks to the electron-hole symmetry.

When the bias voltage is smaller than the gap, eV < ∆, there is only one process contributing to the emission noise, which is shown on panel (a). Electrons in the normal metal with energies in the interval [0, eV ] may hop to the Majorana state by emitting a photon of energy Ω = eV (see Fig. 2a). Since the width of the Majorana peak in the density of states is extremely small, and the DOS of the normal metal is constant, this emission process only shows a negligible dependence on the frequency.

For voltages larger than the gap, another emission process kicks in: electrons from the normal metal can be transferred to the empty states above the TS gap, and emit a photon with Ω in the range [0, eV -∆] (see Fig. 2b). The frequency dependence of this process will reflect the energy dependence of the TS density of states above the gap.

Absorption processes similar to the emission processes discussed above can also be realized. These are shown on panels (c) and (d) of Fig. 2, and are expected to show similar frequency dependence as their emission counterparts. In the case of panel (c), an electron from the normal metal absorbs a photon to be transmitted to the Majorana state. Panel (d) is the equivalent of panel (b) for photon absorption: an electron from the normal metal absorbs a photon and is transmitted to an empty state above the TS gap. Note however that while processes (a) and (c) exist for any voltages, the emission process of panel (b) can only occur for voltages beyond the gap.

Finally, the absorption of photons also allows processes where an electron is transmitted from the topological superconductor to the normal metal, for frequencies Ω ∈ [eV, +∞]. Panels (e) and (f) of Fig. 2 show the corresponding processes, where the TS electron is transmitted to a normal metal empty state, either from the Majorana state [panel (e)] or from the occupied states below the gap [panel (f)].

Results for eV < ∆

We now show the results for the finite frequency emission noise S + (Ω) and absorption noise S -(Ω) obtained at low temperature T e = 0.01∆ for different values of the transparency τ and the voltage V (applied on the normal lead, the TS wire is assumed to be grounded).

The emission noise in the subgap regime eV < ∆ is shown in Fig. 3 (top), at low transparency τ = 0.02 for three different values of the voltage. For each curve, we observe a plateau that extends up to Ω = eV , followed by a sharp drop to zero. This plateau corresponds to the process (a) of Fig. 2, and exists only for frequencies Ω < eV , hence the observed drop. The absorption noise for the same parameters is shown on Fig. 3 (bottom). The most visible feature is a sharp step at Ω = eV , which corresponds to the onset of the process (e) from Fig. 2, whose contribution is independent of frequency.

Smaller structures can be observed at frequency Ω = ∆-eV and Ω = ∆+eV , corresponding respectively to the onset of processes (d) and (f) from Fig. 2. Interestingly, a fourth process also contributes to the absorption noise, namely process (c). However, the latter cannot be associated with any visible feature as it leads to a constant contribution for all positive frequency Ω, thus resulting in a finite background contribution to the absorption noise.

Results for eV > ∆

We now turn to the low-temperature results for the noise at voltages eV > ∆ and in the same low transparency regime (τ = 0.02).

The emission noise S + (Ω) is shown in Fig. 4 (top), for three different voltages beyond the gap. As in the subgap regime, the main feature of the emission noise is the presence of a well-defined plateau extending from Ω = eV -∆ all the way up to Ω = eV , which ends with a sharp drop to zero. Again, this plateau is fully compatible with the process shown in Fig. 2a, as it is expected to lead to a constant contribution, and can only occur for frequencies Ω < eV (since there are no occupied electronic states at higher energy). While this contribution is still present at very low frequency, it is supplemented by another one arising from the process (b) (see Fig. 2) leading to a new structure in the noise. Indeed, for frequencies Ω < eV -∆, electrons close to the metallic Fermi level can be transfered to the empty states above the TS gap while emitting a low-energy photon. The resulting frequency dependence of the emission noise is directly related to the energy dependence of the TS density of states above the gap.

The absorption noise for eV > ∆ is shown on Fig. 4 (bottom). The main feature is again a sharp step for Ω = eV , associated with the process presented in panel (e) of Fig. 2. A smaller structure is visible at Ω = ∆ + eV , and can be directly tied to the onset of process (f). Processes (c) and (d) also contribute to the ab- sorption noise, being present at all frequencies and leading respectively to a constant background contribution and to a steady increase of the noise.

�� � -�Δ �� � �-�Δ �� � �=� ����Δ �� � �= ����Δ �� � � =�����Δ �� � �-�Δ
It follows from the inspection of the finite-frequency noise, for voltages both above and below the TS gap, that the presence of a Majorana bound state leads to a plateau in the emission noise as a function of frequency, accompanied by a sharp drop to zero at Ω = eV .

At this point, it is interesting to relax the constraint of low transparency and study the impact of a higher value of τ . This is shown in Fig. 5, where an intermediate value of the transparency, τ = 0.4, has been used.

One can readily see that increasing the transparency tends to smooth out all the structures visible in the noise and discussed above. Further increasing the transparency and ultimately approaching τ = 1 (not shown) makes all features in the noise spectrum disappear. The best opportunity to identify the noise plateau associated with the Majorana bound state is thus to focus on the case of tunnel junctions.

�� � -�Δ �� � �-�Δ �� � �-�Δ �� � �=� ����Δ �� � =� ����Δ �� � �=� ����Δ 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 Frequency Ω Re [ S + ( Ω ) ] �� � �= �����Δ �� � �= �����Δ �� � �=� ����Δ �� � +�Δ �� � �+�Δ �� � �+�Δ 0.0 0. 5 1.0 1.5 2.0 2.5 3

D. Measurable noise

We end this study with a short discussion of the prediction of the measurable noise. To this end, we choose the inductive coupling scheme introduced in Ref. 20 which involves an LC circuit in the vicinity of the NTS junction (see Fig. 1). This protocol relies on repeated measurements of the charge on the plates of the capacitor of the LC resonant circuit. We aim at investigating the results of the measurable noise S meas (Ω) as a function of the frequency Ω, as defined in Eq. ( 52), which corresponds to the measurable response of this resonant circuit.

Following our investigations on the emission and absorption noise, we focus here on the low transparency regime, as this is the most likely candidate to observe meaningful signatures of the Majorana bound state. The new crucial parameter which dramatically influences the measurable noise is the temperature T LC of the detector.

In the low temperature regime for the detector, T LC Ω, the measurable noise, presented in Fig. 6 for voltages beyond the gap, closely resembles the emission noise obtained earlier for the same set of parameters, showing in particular the characteristic low-frequency behavior of the noise which we could associate with the presence of a Majorana bound state. This is easily understood from the very definition of the measurable noise, Eq. ( 52). Indeed, in this low temperature regime, the Bose-Einstein distribution becomes vanishingly small, so that the measurable noise reduces to

�� � -�Δ �� � �-�Δ �� � �-�Δ �� � �=� ����Δ �� � �=� ����Δ �� � � =�����Δ 0.0 0.5 1.0 1. 5
S meas (Ω T LC ) KS + (Ω) . (56)
As one increases the detector temperature (see Fig. 7 obtained for T LC = ∆), this connection with the emission noise is progressively altered, ultimately leading to the disappearance of any meaningful signatures. However, the typical onsets of the various processes stay unaffected leading to structures in the measurable noise at frequencies Ω = eV -∆ and Ω = eV . In particular, the measurable noise shows a sudden change of sign near Ω = eV , a feature inherited from the sharp drop in the emission noise, which is also directly connected to the presence of the Majorana peak in the TS DOS. It follows that even in this high temperature regime of the detector, the measurable noise can still be used as a tool to uncover signatures of the Majorana bound state.

VI. COMPARISON WITH A N-DOT-S SYSTEM

A remaining question is to inquire whether a nontopological superconductor/normal metal junction bearing zero-energy Andreev bound states (ABS), which also exhibits a zero bias peak in the differential conductance, is likely to produce the same finite-frequency noise characteristics as the NTS junction, which constitutes the focus of this study. We thus consider here a system where a standard BCS superconductor is connected to a normal lead via a single quantum dot. We compute the finite frequency-noise in a regime where the quantum dot has signatures close to a Majorana bound state when looking at the conductance only. This will allow us to point out the unique information obtained by considering the finite-frequency noise, which permits to make a clear distinction between a real Majorana bound state in a topological superconductor and the presence of an accidental bound state at zero energy close to the junction. The Hamiltonian of the system is:

H = H D + j=S,N H j + H T (57)
where H D is the Hamiltonian of the dot level at energy :

H D = σ=↑,↓ d † σ d σ , (58)
H j (j = S, N ) are the Hamiltonians of the BCS superconductor and of the normal lead:

H j = k Ψ † j,k (ξ k σ z + ∆ j σ x ) Ψ j,k (59)
where ∆ S = ∆ is the superconducting gap, and ∆ N = 0. Ψ j,k is a Nambu spinor: andσ x and σ z are Pauli matrices in Nambu space. Finally H T is the tunneling Hamiltonian between the leads and the dots.

Ψ j,k = ψ j,k,↑ ψ † j,-k,↓ T , (60)
ξ k = k 2 /(2m) -µ,
H T = jk Ψ † jk T j (t)d + H.c. ( 61
)
where

d = (d ↑ d † ↓ )
T is a Nambu spinor for the dot electrons. T j is the tunneling amplitude between lead j and the dot, where the constant voltage V j has been included via a Peierls substitution :

T j (t) = t j σ z exp(iσ z V j t)
(62) The tunneling rate between the dot and lead j is further defined as Γ j = πν 0 t 2 j , where ν 0 is the density of states of the metal at Fermi energy.

Calculation of the average current and current correlations is carried out using the Keldysh Green function formalism, and has already been presented for a similar system in Ref. 36. The formula for the mean current to lead j is:

I j = eTr N σ 3 dω 2π Re G r (ω)Σ K (ω) + G K (ω)Σ a (ω)
(63) where the trace is performed in Nambu space, G is the full dot Green function (including the coupling to the leads), and Σ j is the self-energy for lead j. The superscripts r, a, K refer to the retarded, advanced and Keldysh components.

Similarly, the formula for the current autocorrelations at finite frequency Ω is:

S j (Ω) = - e 2 2 Re dω 2π Tr N σ 3 Σ K j G a + Σ r j G K -Σ a j G a + Σ r j G r ω σ 3 Σ K j G a + Σ r j G K + Σ a j G a -Σ r j G r ω+Ω -σ 3 Σ r j G r Σ K j + Σ K j G a Σ a j + Σ r j G K Σ a j -Σ a j G a Σ a j + Σ r j G r Σ r j ω σ 3 G K + G a -G r ω+Ω - e 2 4 Re dω 2π Tr N σ 3 Σ a j -Σ r j -Σ K j ω σ 3 G a -G r + G K ω+Ω + σ 3 G r -G a + G K ω σ 3 Σ r j -Σ a j -Σ K j ω+Ω . (64)
Note that there is an extra term compared to Eq. (51) of Ref. 36, as we are not computing the current cross-correlations between different leads, but instead the autocorrelations. For Ω > 0 (resp. Ω < 0), Eq. (64) gives the emission noise S + (Ω) [resp. the absorption noise S -(Ω)] -to be compared with Eqs. ( 50)-(51) for the NTS case.

We now turn to the results obtained with Eqs. ( 63) and (64). As a first step, we need to specify the param-eters of the N-dot-S system for which the conductance is similar to the one obtained for a NTS junction due to the presence of the Majorana bound state. This means a conductance peak at zero energy, as shown by Eq. ( 54). This can be obtained here if two conditions are satisfied: setting the energy of the dot to zero, and using symmetrical couplings (Γ N = Γ S ). While the first condition is obvious, the second one is illustrated in Fig. 8, which shows the differential conductance G=dI N /dV as a function of the voltage V applied to the normal lead. There, we consider the case of a symmetric junction with Γ N = Γ S = 0.02 (full line), as well as the situation of an asymmetric junction with Γ N = 0.02 and Γ S = 3 Γ N (dotted line). The other parameters are = 0, and 1/(k B T ) = 300, while the superconducting gap ∆ is the unit of energy. One can see that while the symmetric junction leads to a conductance peak which is identical to the one of a NTS junction, the asymmetric one shows a splitting of the conductance peak.

Focusing then on a symmetric junction with a dot at zero energy ( = 0), we now plot the emission noise. Fig. 9 shows S + (Ω) for three values of the voltage below the gap (V = 0.3∆, 0.6∆ and 0.9∆), while Fig. 10 focuses on voltages above the gap (V = 1.2∆, 1.5∆ and 1.8∆), with couplings Γ N = Γ S = 0.02 in both cases. These figures should be compared with their NTS equivalent, Fig. 3 and Fig. 4 respectively.

We see that, as in the case of the NTS junction, there is a clear plateau in the emission noise, which extends up to Ω = eV . This plateau is the manifestation of the discrete level (here the dot level), which is associated with a narrow peak in the density of states. However, an important difference with respect to the NTS case is also visible in Figs. 9 and 10: at small frequency Ω, there is a dip in the emission noise, with a value at Ω = 0 which is exactly half that of the plateau. When probed on a very long timescale (as in the zero frequency noise), the current fluctuations are reduced by a factor 1/2 due to the double-barrier structure with symmetric couplings. When the timescale is reduced to a time smaller than the average transfer time of an electron between an electrode and the dot level (∼ 1/Γ), then the double-barrier nature of the setup has no impact on the current fluctuations, which explains why the dip in S + (Ω) has a width ∼ Γ. This factor 1/2 for the contrast of the observed dip corresponds to the well-known Fano factor reduction in double-barrier symmetric junctions. [START_REF] Martin | Noise in Mesoscopic Physics, Nanophysics : coherence and transport : école d'été de physique des Houches[END_REF] Interestingly, this dip in the emission noise is completely absent in the case of a NTS junction, because the Majorana bound state is an intrinsic part of the superconductor, and is not separated by a barrier. It follows that, unlike the differential conductance, the low-frequency behavior of the emission noise is able to discriminate between the NTS and the N-dot-S system. Granted, our comparison between the noise characteristics of the NTS junction and this system containing a zero energy Andreev bound state is specific to the N-dot-S model we have adopted here. More generally, zero energy Andreev bound states can be generated in the contact region due to disorder and/or to multiple subbands in the nanowire. Studying this more general situation in detail goes well beyond the scope of the present paper. At this point, we cannot absolutely rule out that such an embedded ABS in the superconductor would lead to a frequency noise characteristics similar to that of the NTS junction. We therefore cannot claim that the frequency noise characteristics obtained in this work can uniquely be attributed to the presence of the Majorana fermion in the NTS junction.

VII. CONCLUSION AND PERSPECTIVES

The hunt for Majorana fermions continues to be an active field of research. Besides the zero voltage anomaly which was predicted, and in principle measured, by several experimental and theoretical groups, more signatures of these manifestations of collective electronic excitations are desperately needed.

In this work, we focused on a normal metal/topological superconductor junction subjected to a constant voltage bias using the non-equilibrium Keldysh formalism. Starting from the boundary Green's functions describing the two isolated parts, we were able to express the average current and noise in terms of dressed Green's functions, accounting for tunneling between the leads at all orders. This allows us to investigate the stationary current, the zero frequency noise and, the main result of this analysis, the finite-frequency emission and absorption noise.

Computed for various voltages, at small and intermediate transparency, the finite-frequency noise S + (Ω) and S -(Ω) for the NTS junction show various structures, which can be tied to different emission or absorption processes. In particular, the distinctive feature associated with the presence of the Majorana bound state is a plateau in the noise, extending all the way from Ω = 0 up to a sharp step at frequency Ω = eV for voltages in the subgap regime. The presence of such a plateau is a general feature of a system with a discrete level, which is here the Majorana bound state. However, for a nontopological system with an accidental discrete level at zero-energy, we have shown by computing the emission noise in a N-dot-S system that the double barrier nature of the setup imposes a dip in the plateau at low frequency Ω, which is absent for the NTS junction. In this particu-lar N-dot-S system, the observation of the plateau for the whole frequency range could thus be used to discriminate between an accidental zero-energy Andreev bound state, and a real Majorana bound state. Nevertheless, this still leaves open the possibility that more general superconducting systems which bear ABS embedded in the superconductor would generate the same differential conductance and noise signals as the NTS junction. In any case, tunnel junctions should still be favored, as these characteristic structures are more prominent in the limit of small tunneling amplitudes, and get smoothed out when the transparency is increased.

Finally, we used a generic model for measuring noise consisting of a LC circuit inductively coupled to the normal metal/topological superconductor junction under study. In actual experiments, the temperature of this LC circuit is not necessarily the same as the electronic temperature of the junction. We see that when the temperature of the detection (LC) circuit is comparable to that of the electronic temperature of the NTS junction, we recover the same features as the ones observed in the emission and absorption noise. While higher temperatures of the detection circuit slightly blur the quality of this diagnosis, there still remains specific signatures which can be tied to tunneling processes involving the Majorana bound state. Our results therefore suggest that finitefrequency noise measurements offer a novel diagnostic of the presence of Majorana fermions in condensed matter systems. Extensions could include taking into account the finite length of the TS wire where the two Majorana bound states located at each end hybridize, or changing the chemical potential of the continuum version of the Kitaev chain model used here in order to trigger a transition to the non-topological phase. Using a more realistic model for the topological superconductor, including the electron spin, induced superconductivity, etc. is another notable extension in view of a quantitative comparison with experiments. 37

  00 . Using Dyson's equation for the full advanced and retarded Green's function, Eq. (13), it is convenient to express G

Figure 2 .

 2 Figure 2. Energy diagram of a NTS junction. In medium grey are represented the occupied electronic states and in light grey the empty states. (a)-(b) emission processes, (b) is not available for a bias eV below the gap ∆, (c)-(d) absorption processes involving a transition from the metal to the superconductor, (e)-(f) absorption processes with transitions to the metal. Straight lines correspond to the transfer of electrons while wiggly lines are associated with the absorption or emission of photons.

Figure 3 .

 3 Figure 3. Emission noise S + (Ω) (top) and absorption noise S -(Ω) (bottom) as a function of frequency Ω (in units of ∆) at low transparency (τ = 0.02) computed in the subgap regime |eV | < ∆ for three different values of the voltage bias (eV = 0.3∆, 0.6∆ and 0.9∆), and expressed in units of e 2 ∆/h.

Figure 4 .

 4 Figure 4. Emission noise S + (Ω) (top) and absorption noise S -(Ω) (bottom) as a function of frequency Ω (in units of ∆) at low transparency (τ = 0.02) computed beyond the gap |eV | > ∆ for three different values of the voltage bias (eV = 1.2∆, 1.5∆ and 1.8∆), and expressed in units of e 2 ∆/h.

Figure 5 .

 5 Figure 5. Emission noise S + (Ω) and absorption noise S -(Ω) (in units of e 2 ∆/h) as a function of frequency Ω (in units of ∆) at transparency τ = 0.4 beyond the gap |eV | > ∆, for three different values of the voltage bias (eV = 1.2∆, 1.5∆ and 1.8∆).

Figure 6 .

 6 Figure 6. Measurable noise (in units of e 2 K∆/h) as a function of frequency Ω (in units of ∆) at low transparency (τ = 0.02) for electron and detector temperatures Te = TLC = 0.01∆.

Figure 7 .

 7 Figure 7. Measurable noise (in units of e 2 K∆/h) as a function of frequency Ω (in units of ∆) at low transparency (τ = 0.02) for an electron temperature Te = 0.01∆ and a detector temperature TLC = ∆.

Figure 8 .

 8 Figure 8. Differential conductance G = dIN /dV , for the current to the normal lead IN , in a N-dot-S junction, with symmetric couplings (ΓN = ΓS = 0.02, full line) and asymmetric couplings (ΓN = 0.02, ΓS = 3ΓN , dotted line), as a function of the voltage of the normal lead. Other parameters : = 0, 1/(kBT ) = 300. The superconducting gap ∆ is the unit of energy.

Figure 9 .

 9 Figure 9. Emission noise S + (Ω) for a symmetric N-dot-S junction with ΓN = 0.02, at voltages V = 0.3∆, 0.6∆ and 0.9∆ (indicated by dashed vertical lines). Compare with Fig.3for the case of a NTS junction.

Figure 10 .

 10 Figure 10. Emission noise S + (Ω) for a symmetric N-dot-S junction with ΓN = 0.02, at voltages V = 1.2∆, 1.5∆ and 1.8∆ (indicated by dashed vertical lines). Compare with Fig. 4 for the case of a NTS junction.

ACKNOWLEDGMENTS

The project leading to this publication has received funding from Excellence Initiative of Aix-Marseille University -A*MIDEX, a French "Investissements d'Avenir" program.


Pierre-Jean Malé

Jean-Baptiste Ferdy

Celine Leroy

Olivier Roux

Jérémie Lauth

Arthur Avilez

Alain Dejean

Angélique Quilichini

Jérôme Orivel

Pierre-Jean G Male
email: pjg.male@gmail.com

Jean-Baptiste Ferdy

• Ce

´line Leroy

Je ´re ´mie Lauth

Ange ´lique Quilichini

Je ´ro ˆme Orivel

P.-J G Male

J.-B Ferdy

Á A Avilez Enfa

P.-J G Male

J.-B Ferdy

Á A Avilez

P.-J G Male

Á A Dejean

Á A Quilichini

Á J Orivel

Retaliation in Response to Castration Promotes a Low Level of Virulence in an Ant-Plant Mutualism

Keywords: Evolutionary conflict, Cheater, Overexploitation, Mutualism breakdown, Allomerus decemarticulatus, Hirtella physophora

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Mutualisms are defined as reciprocally beneficial interactions between organisms, but they can also be viewed as 'reciprocal exploitation', thus involving both benefits and costs [START_REF] Herre | The evolution of mutualisms: Exploring the paths between conflict and cooperation[END_REF]. Since natural selection favours individuals that are able to lower their costs and/or increase their benefits, conflicts in resource allocation are expected to cause mutualistic relationships to shift to parasitism [START_REF] Bronstein | The exploitation of mutualisms[END_REF][START_REF] Edwards | Can the failure to punish promote cheating in mutualism?[END_REF]. Striking examples are well-known in the framework of pollination mutualisms. Cheating has evolved in both partners and while some plants produce flowers devoid of nectar, some insects can rob nectar without pollinating their partner [START_REF] Gilbert | Empty flowers[END_REF][START_REF] Inouye | The ecology of nectar robbing[END_REF].

The reproduction of each partner engaged in any horizontally-transmitted mutualism is obviously subjected to a conflict since the resources invested by one partner in its own reproduction are lost for the other partner. Such a trade-off is a major source of instability as, over time, the symbionts may evolve in such a way that they end up completely sterilizing their hosts. Incomplete sterilization is thus considered 'suboptimal virulence' in the literature [START_REF] Jaenike | Suboptimal virulence of an insect-parasitic nematode[END_REF][START_REF] O'keefe | Playing by different rules: The evolution of virulence in sterilizing pathogens[END_REF], which highlights the existence of counterbalancing mechanisms such as the local dispersal of both partners, partner choice and/or retaliation against exploiters [START_REF] Wilkinson | Horizontally acquired mutualisms, an unsolved problem in ecology?[END_REF][START_REF] Szila ´gyi | The evolution of intermediate castration virulence and ant coexistence in a spatially structured environment[END_REF].

Ant-plant mutualisms provide an interesting framework for studying such reproductive trade-offs. Myrmecophytic plants typically provide ants with a nesting space in hollow structures such as thorns or leaf pouches, and sometimes food resources, in exchange for protection against predators and competitors. These relationships are horizontally transmitted and because of this mode of transmission, ants should evolve to be parasitic on their hosts [START_REF] Heil | Protective ant-plant interactions as model systems in ecological and evolutionary research[END_REF][START_REF] Wilkinson | Horizontally acquired mutualisms, an unsolved problem in ecology?[END_REF]. Moreover, because of the obligatory nature of most of these relationships, the plants cannot escape such exploitation by ending the interaction without drastically reducing their fitness. It has been demonstrated in several model systems that the growth of the ant colony is constrained by the nesting space provided by the plant [START_REF] Fonseca | Nesting space limits colony size of the plantant Pseudomyrmex concolor[END_REF][START_REF] Orivel | Dynamics of the association between a long-lived understory myrmecophyte and its specific associated ants[END_REF]. The ants can take advantage of the trade-off between the plant's reproduction and growth and induce gigantism in their host plant by castrating it [START_REF] Frederickson | Conflict over reproduction in an ant-plant symbiosis: Why Allomerus octoarticulatus ants sterilize Cordia nodosa trees[END_REF]. Such a conflict between the reproduction of the plant and the growth of the ant colonies is illustrated by the castration behaviour exhibited by several plant-ants [START_REF] Yu | A castration parasite of an antplant mutualism[END_REF][START_REF] Izzo | Cheating the cheater: Domatia loss minimizes the effects of ant castration in an Amazonian ant-plant[END_REF][START_REF] Gaume | Ant-plant conflicts and a novel case of castration parasitism in a myrmecophyte[END_REF][START_REF] Male | An ant symbiont directly and indirectly limits its host plant's reproductive success[END_REF]. This is, however, based on the major assumption that there is a trade-off between vegetative growth and reproduction in myrmecophytes. Yet, given the modular growth and autotrophy of plants, each module could theoretically be able to pay its own resource costs, for example, through greater photosynthesis in nearby leaves and in some of its floral structures [START_REF] Watson | Developmental constraints: effect on population growth and patterns of resource allocation in a clonal plant[END_REF][START_REF] Obeso | The cost of reproduction in plants[END_REF].

Despite the ant-induced limitations to their reproduction, myrmecophytes manage to reproduce even in the presence of ants. An intermediate level of castration has indeed been observed in associations involving Allomerus plant-ants that probably results from tolerance and/or local dispersal mechanisms [START_REF] Edwards | Tolerating castration by hiding flowers in plain sight[END_REF][START_REF] Izzo | Cheating the cheater: Domatia loss minimizes the effects of ant castration in an Amazonian ant-plant[END_REF][START_REF] Male | An ant symbiont directly and indirectly limits its host plant's reproductive success[END_REF][START_REF] Szila ´gyi | The evolution of intermediate castration virulence and ant coexistence in a spatially structured environment[END_REF]). However, although never studied in these systems, resistance mechanisms are thought to result in the same pattern of intermediate virulence. Indeed, retaliation mechanisms have recently been identified as a key factor in preventing cheating in ant-plant mutualisms, especially in the case of cheating by defection [START_REF] Edwards | Selection for protection in an ant-plant mutualism: host sanctions, host modularity, and the principal-agent game[END_REF][START_REF] Edwards | Can the failure to punish promote cheating in mutualism?[END_REF]. But castration behaviour belongs to another kind of cheating mechanism; namely, overexploitation. In the case of defection, cheating results in a reduction in the partner's benefits, while overexploitation leads to an increase in its costs [START_REF] Douglas | Conflict, cheats and the persistence of symbioses[END_REF][START_REF] Douglas | The symbiotic habit[END_REF]. Moreover, tolerance and resistance mechanisms lead to very different evolutionary paths [START_REF] Best | Resistance is futile but tolerance can explain why parasites do not always castrate their hosts[END_REF]). As a consequence, proving the presence/absence of resistance mechanisms in ant-plants is critical to our understanding of the maintenance of such relationships despite possible exploitation.

In this study, we focus on one of these plant-Allomerus systems involving Hirtella physophora and A. decemarticulatus in which an intermediate level of castration virulence has been demonstrated [START_REF] Male | An ant symbiont directly and indirectly limits its host plant's reproductive success[END_REF]. We aimed to quantify (1) the net benefit of castration for the resident ants in terms of the plant's vegetative growth and thus available nesting space, (2) the plant's response to castration in terms of its investment in myrmecophytic traits (i.e. leaf pouches and extra-floral nectaries), and (3) the putative consequences of any change in the myrmecophytic traits for the ant colony. We experimentally addressed this issue by assessing the plant's investment in myrmecophytic traits when castration is partial, complete or non-existent and analysed the implications of the different plant responses (i.e. gigantism and retaliation) on the continuance of the mutualistic relationship.

Materials and Methods

Study Sites and Model

This study was conducted in French Guiana between August 2008 and December 2011 on two populations of H. physophora less than one kilometre apart in the area of Petit Saut, Sinnamary (05°03 0 30 00 N; 52°58 0 34 00 W) and one population at the Nouragues research station (04°04 0 18 00 N; 52°43 0 57 00 W).

Hirtella physophora is an understory treelet that occurs strictly in pristine Amazonian rainforest. Plant individuals have long-lived leaves that bear a pair of pouches (i.e. domatia), at the base of each lamina [START_REF] Leroy | Comparison between the anatomical and morphological structure of leaf blades and foliar domatia in the ant-plant Hirtella physophora (Chrysobalanaceae)[END_REF]. The domatium results from the curling under of the leaf margin and the tissues show profound morpho-anatomical modifications compared to the leaf lamina [START_REF] Leroy | Comparative structure and ontogeny of the foliar domatia in three neotropical myrmecophytes[END_REF]. Circular extra-floral nectaries (EFNs) occur on the abaxial surface of the lamina and inside the domatia. They differ morphologically, with the EFNs in the domatia being three times larger than the ones on the lamina [START_REF] Leroy | Comparison between the anatomical and morphological structure of leaf blades and foliar domatia in the ant-plant Hirtella physophora (Chrysobalanaceae)[END_REF]. H. physophora is a self-incompatible, entomogamous species and two flowering periods can be distinguished annually: from December to February and June to August (Male ´2011).

In the study area, plant individuals are almost exclusively inhabited by A. decemarticulatus with a single colony per plant [START_REF] Solano | Myrmecophytes of the understory of French Guianian rainforests: Their distribution and their associated ants[END_REF]. Moreover, A. decemarticulatus has never been found in association with another myrmecophyte [START_REF] Grangier | Mechanisms driving the specificity of a myrmecophyteant association[END_REF]. As in any other protective mutualism between ants and plants, A. decemarticulatus workers protect their host plant from defoliators, thus favouring its vegetative growth [START_REF] Grangier | Indirect defense in a highly specific ant-plant mutualism[END_REF][START_REF] Orivel | Dynamics of the association between a long-lived understory myrmecophyte and its specific associated ants[END_REF]). However, A. decemarticulatus also destroys almost two-thirds of the floral buds of its host plant, resulting in very low fruit production [START_REF] Male | An ant symbiont directly and indirectly limits its host plant's reproductive success[END_REF]. The castration behaviour of A. decemarticulatus is thus characterized by an intermediate level of virulence.

Experimental Protocol

A total of 54 flowering H. physophora inhabited by welldeveloped colonies of A. decemarticulatus were randomly assigned to one of three groups of 18 individuals. The plants in these groups were similar in terms of size, base diameter, and number of branches, leaves and inflorescences. Each group was subjected to one of the three following experimental treatments: (1) all of the inflorescences were cut off at an early stage in their development ('total castration'); (2) untreated plants underwent the natural, ant-induced partial castration ('intermediate castration'); and (3) ants were excluded from all of the inflorescences by applying rings of Tanglefoot Ò on either side of the peduncles on shoots previously protected by aluminium foil ('no castration'). At the beginning of the experiment, the youngest leaf on every branch was marked and then, each month, the number of new leaves and inflorescences was recorded. Over the course of the experiment, the plants were subjected to the same experimental treatment again when necessary (i.e. cutting off the inflorescences or excluding ants at an early stage of inflorescence development).

The experiment ended 17 months later, i.e. at the beginning of the fourth flowering period. Nine individuals (i.e. three in the 'total castration' group, four in the 'intermediate castration' group and two in the 'no castration' group) were not taken into account because of the death of their associated ant colony or of the plant itself over the course of the experiment. We recorded the number of remaining pre-treatment leaves. When possible, the two youngest pre-treatment leaves and two leaves that had reached maturity during the experiment were collected to compare morphological characteristics between them. For each leaf collected, we estimated its surface area using ImageJ v1.34 software and quantified the volumes of the two domatia. Domatia volume was estimated based on the volume of an ellipsoid for which the three elliptic radii were equal to the length, width and height of the domatium. We also counted the number of EFNs and measured the surface areas of four of them on the lamina and four others in the domatia.

We then assessed the effect of the experimental treatment on myrmecophytic characteristics by comparing (1) the domatia volumes, (2) the number of EFNs per leaf, and (3) the surface areas of the domatia and lamina EFNs based on the experimental treatment and with pre-treatment leaves. The plants' investment in vegetative parts was assessed by comparing the number of newly-produced leaves and the abscission rate of old leaves between the three groups. We also compared the surface areas of the leaves based on the three experimental treatments and with the pre-treatment leaves. The effect of the experimental treatments on plant investment in reproduction was quantified by comparing the final number of inflorescences between the three experimental groups.

We also sampled the youngest five leaves on one branch for 20 different plants and the penultimate leaf on up to eight branches for 20 other H. physophora individuals inhabited by A. decemarticulatus. The domatia volume and the number and surface area of the domatia EFNs were determined as described above. We counted the number of worker larvae in each domatium and then verified that A. decemarticulatus larvae were preferentially clustered in the penultimate pair of domatia. As a consequence, only the penultimate pair of domatia was used in the subsequent analyses. Since numerous domatia were devoid of larvae, we chose to test separately whether the presence/absence of larvae in the domatia was influenced by domatia volume, and, when larvae were present, whether their number was influenced by domatia volume.

Statistical Analyses

All of the statistical analyses were conducted using R (R Development Core Team 2009). The data were analyzed using generalized linear mixed-effect models (GLMM), most of them conducted with the lme function combined with the Anova function in the nlme and car packages, respectively [START_REF] Pinheiro | nlme: Linear and Nonlinear Mixed Effects Models[END_REF][START_REF] Fox | An R companion to applied regression[END_REF]. Population or plant ID was modeled as a random block effect in all of the analyses. When appropriate, plant or leaf ID was nested in population/plant ID. We assessed model fit through the visual evaluation of residual plots and by using the Shapiro-Wilk normality test when appropriate. We tested for the autocorrelation and heteroskedasticity of the residuals by conducting the Durbin-Watson and Breusch-Pagan tests using, respectively, the dwtest and the bptest functions in the lmtest package [START_REF] Zeileis | Diagnostic checking in regression relationships[END_REF]. When necessary, data were Box-Coxtransformed using the boxcox function in the MASS package [START_REF] Venables | Modern applied statistics with S[END_REF]. Because of their binomial nature, the abscission rate of old leaves and the probability of larval presence in the domatia were analyzed using the lmer function in the lme4 package [START_REF] Bates | lme4: Linear mixed-effects models using S4 classes[END_REF]) combined with the Anova function.

Results

Castrated Plants Produce More Leaves

Non-castrated plants produced fewer leaves than totally or partially castrated ones (Fig. 1; see also electronic supplementary material), but bore as many remaining pre-treatment leaves (GLMM: df = 2; v 2 = 0.067; p = 0.9672). There was, however, no significant difference in leaf production between totally and partially castrated plants (Fig. 1; see also electronic supplementary material). During the flowering period that immediately followed the end of the experiment, the three groups produced the same number of inflorescences (GLMM: df = 2; v 2 = 2.923, p = 0.2319). This result shows that the investment by plants in reproduction was not affected by the outcome of the previous flowering periods.

Castrated Plants Produce Smaller Domatia

The surface area of new leaves did not differ significantly according to the treatment or in comparison with pre-treatment leaves (GLMM: df = 3; v 2 = 3.222, p = 0.3587), but new leaves on totally castrated plants bore significantly smaller domatia than did both pre-treatment leaves and leaves on plants which experienced partial castration by ants (GLMM: df = 3; v 2 = 15.758, p = 0.0013; Fig. 2a; see also electronic supplementary material). Although the experimental treatment did not affect the total number of EFNs per leaf (GLMM: df = 3; v 2 = 1.346, p = 0.7184), totally castrated plants produced leaves with significantly smaller lamina and domatia EFNs than both pre-treatment leaves and new leaves on non-castrated plants (GLMM, lamina EFN: df = 3; v 2 = 9.947, p = 0.0190; domatia EFN: df = 3; v 2 = 7.903, p = 0.0049; Fig. 2b,c; see also electronic supplementary material).

Smaller Domatia Shelter Less Ant Larvae

In non-manipulated plants, worker larvae were preferentially clustered in the domatia of the penultimate leaf (GLMM: df = 4; v 2 = 108.75, p \ 0.0001; Fig. 3; see also electronic supplementary material). The probability of larval presence in the penultimate pair of domatia increased with domatia volume but was influenced neither by the number of domatia EFNs, nor by the mean domatia EFN surface area (Table 1a).

Similarly, when present, the quantity of larvae in the penultimate pair of domatia was also positively influenced by domatia volume, but not by the number of domatia EFNs or by the mean domatia EFN surface area (Table 1b). These results mean that the smaller the domatium, the less chances there are that it shelters any larvae, and if it nevertheless does, the less larvae it shelters.

Discussion

Allomerus ants are known to destroy the inflorescences of their host plant [START_REF] Frederickson | Conflict over reproduction in an ant-plant symbiosis: Why Allomerus octoarticulatus ants sterilize Cordia nodosa trees[END_REF][START_REF] Male | An ant symbiont directly and indirectly limits its host plant's reproductive success[END_REF][START_REF] Edwards | Tolerating castration by hiding flowers in plain sight[END_REF]. Our results demonstrate that this castration behaviour induces an increase in the H. physophora growth rate: castrated plants produce more leaves than those bearing inflorescences from which ants have been excluded. This confirms the idea that A. decemarticulatus colonies obtain more living space than putative non-castrating ants thanks to their castrating behaviour. By castrating their host plant, A. decemarticulatus colonies thus exploit the trade-off between vegetative growth and reproduction in H. physophora. Consequently, the destruction of the plant's sexual organs can be assimilated to parasitic castration inducing gigantism [START_REF] Hall | Parasitic castration: A perspective from a model of dynamic energy budgets[END_REF] which has already been observed in another ant-plant interaction [START_REF] Frederickson | Conflict over reproduction in an ant-plant symbiosis: Why Allomerus octoarticulatus ants sterilize Cordia nodosa trees[END_REF].

We show that the total castration of H. physophora induces a decrease in its investment in myrmecophytic traits. This reduction affects both housing and food resources with a decrease in the domatia volume and a reduction in the surface areas of the EFNs, but does not affect non-myrmecophytic traits such as surface area of leaves. The loss of space resulting from the decrease in the volume of the domatia is likely to directly affect the development and fitness of too virulent colonies, as nesting space is a major limiting factor to the growth and reproduction of the A. decemarticulatus colonies [START_REF] Orivel | Dynamics of the association between a long-lived understory myrmecophyte and its specific associated ants[END_REF]. Our results demonstrate that a reduction in the investment in myrmecophytic traits affects the newly-produced domatia that are, under natural conditions, the ones that host most of the ant brood. Moreover, the number of larvae that are hosted in a domatium depends on the volume of that domatium. As a consequence, the development of a putative fully-castrating ant colony is very likely to be slower than that of a partially-castrating ant colony. This slowing down in colony growth should directly translate into a loss of fitness because the production of alates in A. decemarticulatus has been shown to be positively correlated to the number of workers [START_REF] Orivel | Dynamics of the association between a long-lived understory myrmecophyte and its specific associated ants[END_REF].

One still might wonder if the decrease in domatia volume can be compensated by an increase in the number of domatia. Because the probability of sheltering larvae, and not only the number of larvae, is also negatively correlated to domatia volume, small domatia can be completely lost for the ant colony. As a consequence, the cost of total castration could be greater in terms of a loss in fitness for the ant colony than the net benefit of more leaves, i.e. nesting units, produced. Moreover, the cost of total castration also affects the size of the EFNs located inside the domatia. Since the quantity of nectar produced is proportional to the volume of the nectarial parenchyma Letters indicate significant differences (GLMM, p values \0.05) [START_REF] Pacini | Nectar biodiversity: A short review[END_REF], the smaller the EFNs, the less nectar they are believed to produce. Ants inhabiting totally castrated plants should thus experience a potentially harmful decrease in their carbohydrate intake which might potentially affect the performance of the colony [START_REF] Dussutour | Ant workers die young and colonies collapse when fed a high-protein diet[END_REF]. By analogy with plant-rhizobia interactions, both a reduction in living space and in the amount of food provided can be considered retaliation by the plant [START_REF] West | Sanctions and mutualism stability: Why do rhizobia fix nitrogen[END_REF]. Such retaliation mechanisms against castrating parasites are shown here for the first time in the framework of ant-plant interactions.

The destruction of a fraction of the sexual organs seems to be the best compromise because it maximizes the costto-benefit ratio for both partners. Retaliation was only detected when the virulence of the castration was at its maximum. As a consequence, over the long term, it is unlikely that ants that totally castrate their host plant gain a fitness advantage compared to ants with an intermediate level of virulence. Nevertheless, intermediate castration virulence is not an optimum situation for H. physophora. Plants with a lower tolerance to castration should obtain a higher fitness benefit. But the global fitness of an individual has to be considered over its lifespan. Indeed, [START_REF] Palmer | Synergy of multiple partners, including freeloaders, increases host fitness in a multispecies mutualism[END_REF] demonstrated that the provisional association with sterilizing ants enhanced the lifetime fitness of Acacia drepanolobium. As the energy that a plant invests in reproduction is lost for vegetative investment, there is a trade-off between reproduction and survival, leading to a trade-off between current reproductive output and future reproduction [START_REF] Obeso | The cost of reproduction in plants[END_REF]. The ant-induced reduction in one reproductive event is thus likely to result in an increase in the life expectancy of the plant, and, consequently, in an increase in the plant's subsequent reproductive success.

Fig. 1 Fig. 2

 12 Fig.1Barplot of the mean (± SE) number of leaves produced during the experiment as a function of the treatment ('No castration', plants with inflorescences from which ants were excluded; 'Intermediate castration', untreated plants experiencing natural, ant-induced partial castration; 'Total castration', plants with all of their inflorescences experimentally removed at an early stage). Population ID was taken into account as a random effect. Letters indicate significant differences in the number of leaves (GLMM, p values \0.05)

Fig. 3

 3 Fig.3Barplot of the mean (± SE) number of larvae sheltered in the domatia as a function of the leaf location ('Leaf one', the youngest leaf on the branch; the other leaves are numbered subsequently). Letters indicate significant differences (GLMM, p values \0.05)

Table 1

 1 Effect of domatia volume, number of domatia EFNs and mean diameter of domatia EFNs on (a) presence/absence of larvae in domatia, and (b) number of larvae when present in domatia DF number of degrees of freedom, SE standard error * Statistical significance at the 5 % level, ** Statistical significance at the 1 % level

	Estimate	SE	DF	v 2	p value

Acknowledgments

We are grateful to the Laboratoire Environnement de Petit Saut and the Nouragues scientific station for furnishing logistical help, to Dr. Jacqui Shykoff and Pr. Doyle McKey for insightful comments and to Andrea Yockey-Dejean for proofreading the manuscript. Financial support for this study was provided by a research program of the French Agence Nationale de la Recherche (research agreement n°ANR-06-JCJC-0109-01), by the ESF-EURO-CORES/TECT/BIOCONTRACT program, by the Fondation pour la Recherche sur la Biodiversite ´(research agreement n°AAP-IN-2009-050), by the Programme Convergence 2007-2013 Re ´gion Guyane from the European Community, and by the Programme Amazonie II of the French Centre National de la Recherche Scientifique. This work has benefited from ''Investissement d'Avenir'' grants managed by the Agence Nationale de la Recherche (CEBA, ref. ANR-10-LABX-25-01 and TULIP, ref. ANR -10-LABX-0041).