Jérôme Rech

Dario Ferraro

Thibaut Jonckheere

Luca Vannucci

Maura Sassetti

Thierry Martin

Minimal Excitations in the Fractional Quantum Hall Regime

Keywords: numbers: 73.23.-b, 42.50.-p, 71.10.Pm, 72.70.+m, 73.43.Cd

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

Because of its potential application to quantum information processing, time-dependent quantum transport in open coherent nanostructures attracts prodigious attention. Recent years have seen the emergence of several attempts to manipulate elementary charges in quantum conductors [1][2][3][4]. This opened the way to the field of electron quantum optics (EQO) [5] characterized by the preparation, manipulation and measurement of singleparticle excitations in ballistic conductors.

In this context, levitons -the time-resolved minimal excitation states of a Fermi sea -were recently created and detected in two-dimensional electron gas [4,6], 20 years after being theoretically proposed [7][8][9]. These many-body states are characterized by a single particle excited above Fermi level, devoid of accompanying particle-hole pairs [10]. The generation of levitons via voltage pulses does not require delicate circuitry and has thus been put forward as a solid candidate for quantum bit applications, in particular the realization of electron flying qubits [11,12].

Interaction and quantum fluctuations strongly affect low dimensional systems leading to dramatic effects like spin-charge separation and fractionalization [13][14][START_REF] Giamarchi | Quantum Physics in One Dimension[END_REF]. These remarkable features were investigated by looking at both time-resolved current [16][17][18] and noise measurements [19][20][21][22][23]. While the emergence of many-body physics and the inclusion of interactions [24][25][26][27] was recently addressed in the framework of EQO, a conceptual gap still remains when it comes to generating minimal excitations. This is particularly true when the groundstate is a strongly correlated state, as are the edge channels of a fractional quantum Hall (FQH) system [28], a situation which has remained largely unexplored so far for time-dependent drives [29]. The building blocks of such chiral conductors are no longer electrons but instead anyons, which have a fractional charge and statistics [30]. For Laughlin filling factors [31], these anyons are Abelian quasiparticles, but more exotic situations involving non-Abelian anyons [32] are predicted. Our understanding of these nontrivial objects would benefit from being able to excite only few anyons at a time [33], allowing us to study their transport and exchange properties, and to combine them through interferometric setups. This calls for the characterization of minimal excitations in the FQH regime.

I B V (t) FQHE FQHE 1 4 3 2
In this letter, we study levitons in the edge channels of the fractional quantum Hall regime by analyzing the partition noise at the output of a quantum point contact (QPC). Our results rely on a dual approach combining perturbative and exact calculations of the noise in a Hanbury-Brown and Twiss (HBT) [34,35] configuration. We also provide results in the time-domain, investigating leviton collisions with Hong-Ou-Mandel (HOM) [4,36] interferometry.

Consider a FQH bar (see Fig. 1) with Laughlin filling factor ν = 1/(2n + 1) (n ∈ N), described in terms of a hydrodynamical model [37] by the Hamiltonian ( = 1)

H = v F 4π dx   µ=R,L (∂ x φ µ ) 2 - 2e √ ν v F V (x, t) ∂ x φ R   , (1)
where the bosonic fields φ R,L propagate along the edge with velocity v F and are related to the quasiparticle annihilation operator as

ψ R,L (x) = U R,L √ 2πa e ±ik F x e -i √ νφ R,L (x)
(with a a cutoff parameter and U a Klein factor), and V (x, t) is an external potential applied to the upper edge at contact 1.

Working out the equation of motion for the field φ R ,

(∂ t + v F ∂ x ) φ R (x, t) = e √ νV (x, t)
, one can relate it to the unbiased case using the transformation

φ R (x, t) = φ (0) R (x, t) + e √ ν t -∞ dt V (x , t ), (2)
with x = x -v F (t -t ), and φ

(0) R is the free chi- ral field, φ (0) R (x, t) = φ (0)
R (x -vt, 0). Focusing first on the regime of weak backscattering (WB), the tunneling Hamiltonian describing the scattering between counter-propagating edges at the QPC can be written, in terms of the transformed fields, Eq. ( 2), as

H T = Γ(t)ψ † R (0)ψ L (0) + H.c., where we introduced Γ(t) = Γ 0 exp ie * t -∞ dt V (t ) [38],
with the bare tunneling constant Γ 0 , the fractional charge e * = νe and assuming a voltage V (t) applied over a long contact, in accordance with the experimental setup [4], allowing us to simplify

t -∞ dt V (v F (t -t), t ) t -∞ dt V (t ).
The applied time-dependent voltage consists of an AC and a DC part V (t) = V dc + V ac (t), where by definition V ac averages to zero over one period T = 2π/Ω. The DC part indicates the amount of charge propagating along the edge due to the drive. The total excited charge Q over one period is then:

Q = T 0 dt I(t) = ν e 2 2π T 0 dtV (t) = qe, (3)
where the fractional conductance quantum is G 0 = νe 2 /2π and the number of electrons per pulse is q = e * V dc Ω . The AC voltage generates the accumulated phase experienced by the quasiparticles ϕ(t) = e * t -∞ dt V ac (t ), characterized by the Fourier components p l of e -iϕ (t) .

In a 1D Fermi liquid, the number of electron-hole excitations resulting from an applied time-dependent voltage bias is connected to the current noise created by the pulse scattering on a QPC [7,9,[START_REF] Lee | Orthogonality catastrophe in a mesoscopic conductor due to a time-dependent flux[END_REF] which acts as a beamsplitter, as in a HBT setup [34,35]. For FQH edge states however, scattering at the QPC is strongly non-linear as it is affected by interactions. Special care is thus needed for the treatment of the point contact, and the definition of the excess noise giving access to the number of excitations.

The quantity of interest is the photo-assisted shot noise (PASN), i.e. the zero-frequency current noise measured from contact 3, and defined as

S = 2 dτ T 0 d t T δI 3 ( t + τ 2 )δI 3 ( t - τ 2 ) ( 4
)
where δI 3 (t) = I 3 (t)-I 3 (t) and the output current I 3 (t) reduces, since contact 2 is grounded, to the backscattered

S 0 = 2 T e * Γ 0 v F 2 Ω Λ 2ν-2
as a function of the number of electron per pulse q, for different reduced temperatures θ and filling factor ν = 1/3, in the case of a square (bottom), a cosine (middle) and a periodic Lorentzian drive with half-width at half-maximum η = W/T = 0.1 (top).

current I B (t), readily obtained from the tunnel Hamiltonian

I B (t) = ie * Γ(t)ψ † R (0, t)ψ L (0, t) -H.c. . (5)
When conditions for minimal excitations are achieved in the perturbative regime, excitations should be transmitted independently, leading to Poissonian noise. It is thus natural to characterize minimal excitations as those giving a vanishing excess noise at zero temperature:

∆S = S -2e * I B (t) , (6)
where I B (t) is the backscattered current averaged over one period.

Using the zero-temperature bosonic correlation func-

tion φ R/L (τ )φ R/L (0) c = -log (1 + iΛτ ), this excess noise is computed perturbatively up to order Γ 2 0 , yield- ing [38] ∆S = 2 T e * Γ 0 v F 2 Ω Λ 2ν-2 1 Γ(2ν) × l P l |l + q| 2ν-1 [1 -Sgn (l + q)] , (7)
where Λ = v F /a is a high-energy cutoff and

P l = |p l | 2
is the probability for a quasiparticle to absorb (l > 0) or emit (l < 0) l photons, which depends on the considered drive [38]. These probabilities P l also depend on q, as the AC and DC components of the voltage are not independent. Indeed, we are interested here in a periodic voltage V (t) consisting of a series of identical pulses, with V (t) close to 0 near the beginning and the end of each period. This implies that the AC amplitude is close to the DC one. Our formalism could also be used to perform a more general analysis by changing these contributions independently. In particular, fixing the DC voltage and changing the AC amplitude allows us to perform a spectroscopy of the probabilities themselves. Conversely, changing the DC voltage at fixed AC amplitudes, we can reconstruct the tunneling rate associated with each photo-assisted process [10] in the same spirit as finite frequency noise calculations [START_REF] Ferraro | [END_REF]. However, this broader phenomenology does not provide any additional information concerning the possibility of creating minimal excitation by applying periodic pulses. In Fig. 2, we show the variation of the excess noise as a function of q, for several external drives at ν = 1/3 and various reduced temperatures θ = k B Θ/Ω (Θ the electronic temperature). At θ = 0, only the periodic Lorentzian drive leads to a vanishing excess noise, and only for integer values of q. This confirms that as in the 1D Fermi liquid, and as mentioned in earlier work [9], optimal pulses have a quantized flux and correspond to Lorentzians of area dtV = m2π/e * (with m an integer number of fractional flux quanta). More intriguingly, however, this vanishing of ∆S occurs for specific values of q: while levitons in the FQH are also minimal excitations, they do not carry a fractional charge and instead correspond to an integer number of electrons. This shows that integer levitons are minimal excitation states even in the presence of strong electron-electron interactions, and that it is not possible to excite individual fractional quasiparticles using a properly quantized Lorentzian voltage pulse in time. Indeed, it is easy to note that, under these conditions, at q = ν (single quasiparticle charge pulse) no specific feature appears in the noise and ∆S = 0. While fractional minimal excitations may exist, they cannot be generated using either Lorentzian, sine or square voltage drives.

Close to integer q the behavior of ∆S is strongly asymmetric. While a slightly larger than integer value leads to vanishingly small excess noise, a slightly lower one produces a seemingly diverging contribution. Indeed, exciting less than a full electronic charge produces a strong disturbance of the ground state, and ultimately leads to the generation of infinitely many particle-hole excitations, which is reminiscent of the orthogonality catastrophe [7,[START_REF] Lee | Orthogonality catastrophe in a mesoscopic conductor due to a time-dependent flux[END_REF]41].

For comparison with experiments, we compute the excess noise at θ = 0. This calls for a modified definition of ∆S (in order to discard thermal excitations):

∆S = S -2e * I B (t) coth q 2θ , (8)
which coincides with Eq. ( 6) in the θ → 0 limit. The finite temperature results (see Fig. 2) cure some inherent limitations of the perturbative treatment at θ = 0 (diverging behavior close to integer q). The noiseless status of the Lorentzian drive is confirmed, as ∆S 0 at low enough temperature for some values of q (yet shifted compared to the θ = 0 ones).

Our perturbative analysis is valid when the differential conductance is smaller than G 0 . This condition can be achieved on average (Γ 0 is then bounded from above), but it is not fulfilled in general when the voltage drops near zero because of known divergences at zero temperature. In order to go beyond this WB picture, we now turn to an exact non-perturbative approach for the special filling ν = 1/2. While this case does not correspond to an incompressible quantum Hall state, it nevertheless provides important insights concerning the behavior of physical values of ν beyond the WB regime. The agreement between the two methods in the regime where both are valid makes our results trustworthy.

We thus extend the refermionization approach for filling factor ν = 1/2 [42,43] to a generic AC drive [38]. Starting from the full Hamiltonian expressed in terms of bosonic fields, one can now write the tunneling contribution introducing a new fermionic entity, ψ(x, t) ∝ e i(φ R (x,t)+φ L (x,t))/ √ 2 . Solving the equation of motion for ψ(x, t) near x = 0, one can define a relation between this new field taken before (ψ b ) and after (ψ a ) the QPC

ψ a (t) = ψ b (t) -γΩe iϕ(t)+iqΩt t -∞ dt e -γΩ(t-t ) × e -iϕ(t )-iqΩt ψ b (t ) -H.c. , (9)
allowing us to treat the scattering at the QPC at all orders. Expressing the current and noise in terms of ψ a and ψ b , and using the standard correlation function ψ † b (t)ψ b (t ) = dω 2πv F e iω(t-t ) f (ω) (with f the Fermi function), we derive an exact solution for both the backscattered current and PASN. As the DC noise at a QPC does not remain Poissonian when its transmission increases, our definition of ∆S is further extended to treat the non-perturbative regime. In the ν = 1 case, where an exact solution exists, it is standard to compare the PASN to its equivalent DC counterpart [4,9] obtained with the same V dc , and V ac = 0. Here, in order to account for the nontrivial physics involved at the QPC in the FQH, it makes sense to compare our PASN to the DC noise which one obtains for the same charge transferred at the QPC, over one period of the AC drive. At zero temperature, ∆S is redefined as:

∆S = S -2e * I B + (e * ) 2 T 2γ sin T γe * I B , (10)
where e * = νe = e/2, and γ = |Γ0| 2 πav F Ω is the dimensionless tunneling parameter. This definition coincides with the Poissonian one Eq. ( 6) at low γ, in that it vanishes for the same values of q.

Results for ∆S at ν = 1/2 are presented in Fig. 3. At low γ, structures appear as a function of q, which are very similar to the perturbative calculation (Fig. 2). For the Lorentzian drive only, the excess noise approaches zero close to integer values of q in the tunneling regime γ 1. When increasing γ the position of these minima gets shifted and the excess noise eventually becomes featureless, independently of the AC drive. In the γ → +∞ limit (not shown) the Lorentzian drive shows signatures of Poissonian electron tunneling at the QPC occurring at q multiples of ν, consistent with the duality property of the FQH regime [42]. This Poissonian behavior, not observed for other drives, is also confirmed by the strong backscattering perturbative treatment. At finite temperature, our results are almost unaffected for θ γ, while larger temperatures tend to smear any variations in q.

Levitons can also be explored in the time domain through electronic Hong-Ou-Mandel (HOM) interferometry [4,6]. By driving both incoming channels, one can study the collision of synchronized excitations onto a beam-splitter, as two-particle interferences reduce the current noise at the output, leading to a Pauli-like dip. Fig. 4 shows the normalized HOM noise ∆Q [24] as a function of the time delay τ between applied drives. While this does not constitute a diagnosis for minimal excitations, it reveals the special nature of levitons in the WB regime, as the normalized HOM noise is independent of temperature and filling factor [38,44], reducing at q = 1 to the universal form:

∆Q(τ ) = sin 2 πτ T sin 2 πτ T + sinh 2 (2πη) . ( 11
)
The same universal behavior is also obtained for fractional q = ν in the strong backscattering regime (tunneling of electrons at the QPC). Interestingly, although the HOM noise and the PASN are very different from their Fermi liquid counterparts, an identical expression for ∆Q(τ ) was also obtained in this case [10] (where it is viewed as the overlap of leviton wavepackets). Finally, in addition to the excess noise, the timeaveraged backscattering current I B (t) also bears peculiar features. In contrast to the Ohmic behavior observed in the Fermi liquid case, I B (t) shows large dips for integer values of q (see Fig. 5). These dips are present for all types of periodic drives, and cannot be used to detect minimal excitations. However, the spacing between these dips provides an alternative diagnosis (from DC shot noise [19,20]) to access the fractional charge e * of Laughlin quasiparticles, as q is known from the drive frequency and the amplitude V dc [45].

Real-time quasiparticle wave packet emission has thus been studied in a strongly correlated system, showing the existence of minimal excitations (levitons) in edge states of the FQH. These occur when applying a periodic Lorentzian drive with quantized flux, and can be detected as they produce Poissonian noise at the output of a Hanbury-Brown and Twiss setup in the weak backscattering regime. Although FQH quasiparticles typically carry a fractional charge, the charge of these noiseless excitations generated through Lorentzian voltage pulse corresponds to an integer number of e. Furthermore, our findings are confirmed for arbitrary tunneling using an exact refermionization scheme. Remarkably enough, in spite of the strong interaction, two FQH leviton collisions bear a universal Hong-Ou-Mandel signature identical to their Fermi liquid analog. Possible extensions of this work could address more involved interferometry of minimal excitations as well as their generalization to non-Abelian states.

Note added in proofs: During the completion of this work, it came to our attention that a simple argument can rule out the possibility of minimal excitations beyond the results presented here. Starting from Eq. ( 7), one readily sees that a minimal excitation (∆S = 0) can only be realized if P l = 0 for all l ≤ -q, independently of the filling factor. At ν = 1, it was shown [7][8][9] that minimal excitations were associated with quantized Lorentzian pulses, so that this type of drive is the only one satisfying the constraint of vanishing P l . Since this condition is independent of ν, it follows that also at fractional filling, minimal excitations can only be generated using Lorentzian drives with quantized charge q ∈ Z.

quasiparticle annihilation operators ψ µ (x, t) (µ = R, L) through the bosonization identity

ψ R/L (x, t) = U R/L √ 2πa e ±ik F x e -i √ νφ R/L (x,t) , ( B4
)
where a is a short distance cutoff and U µ are Klein factors.

Focusing first on the case where no QPC is present, one can derive the following equations of motion for the bosonic fields

(∂ t + v F ∂ x ) φ R (x, t) = e √ νV (x, t) (B5) (∂ t + v F ∂ x ) φ L (x, t) = 0 (B6)
It follows that the effect of the external voltage bias can be accounted for by a rescaling of the right-moving bosonic field φ R (x, t) = φ

(0) R (x, t) + e √ ν t -∞ dt V (x , t ) (with φ (0)
R the solution in absence of time dependent voltage) or alternatively by a phase shift of the quasiparticle operator of the form

ψ R (x, t) -→ ψ R (x, t) e -iνe t -∞ dt V (x ,t ) (B7) where x = x -v F (t -t ).
Accounting for this phase shift, the tunneling Hamiltonian which describes the scattering of single quasiparticles at the QPC (x = 0) in the weak backscattering regime is given by

H T = Γ 0 exp iνe t -∞ dt V (v F (t -t), t ) ψ † R (0)ψ L (0) + H. c
., with the bare tunneling constant Γ 0 .

Considering for simplicity the experimentally motivated situation of a long contact, located at a distance d from the QPC, one can write the bias voltage as V (x, t) = V (t)θ (-x -d) where V (t) is a periodic timedependent voltage. The tunneling Hamiltonian can then be simplified as

H T = Γ 0 exp iνe t-d v F -∞ dt V (t ) ψ † R (0)ψ L (0) + H.c. (B8)
Note that the time delay d/v F can safely be discarded as it corresponds to a trivial constant shift in time of the external drive. The backscattering current is readily obtained from H T , after defining Γ(t) = Γ 0 exp ie * t -∞ dt V (t ) and e * = νe:

I B (t) = ie * Γ(t)ψ † R (0, t)ψ L (0, t) -H.c. . (B9)
Expanding to order Γ 2 0 and taking the average over one period, the backscattering current becomes

I B (t) = - 2ie * T Γ 0 2πa 2 +∞ -∞ dτ e 2νG(-τ ) × T 0 d t sin e * t+ τ 2 t-τ 2 dt V (t ) , (B10)
with G(t -t ), the connected bosonic Green's function G(t -t ) = φ µ (0, t)φ µ (0, t ) c , which reads at zero temperature G(t -t ) = -log 1 + i v F (t-t ) a . The unsymmetrized shot noise is written in terms of I B (t) [45] as S(t, t ) = I B (t)I B (t ) -I B (t) I B (t ) . To second order Γ 2 0 , the zero-frequency time-averaged shot noise becomes:

S = 2 dτ T 0 d t T S t + τ 2 ; t - τ 2 = e * Γ 0 πa 2 dτ e 2νG(-τ ) × T 0 d t T cos e * t+ τ 2 t-τ 2 dt V (t ) . (B11)
The excess shot noise at zero temperature then takes the form

∆S = S -2e * I B (t) = e * Γ 0 πa 2 dτ T 0 d t T exp [2νG (-τ )] × exp ie * t+ τ 2 t-τ 2 dt V (t ) . (B12)
Splitting the voltage into its DC and AC part, and using the Fourier coefficients p l introduced in Eq. (A1), one has

∆S = e * Γ 0 πa 2 l |p l | 2 dτ e i(l+q)Ωτ +2νG(-τ ) = 2 T e * Γ 0 v F 2 1 Γ(2ν) Ω Λ 2ν-2 × l P l |l + q| 2ν-1 [1 -Sgn (l + q)] , (B13)
where the zero-temperature expression of G(-τ ) has been used, which allows to perform the integration. As in the main text, we also introduced the notations q = e * V dc Ω for the charge per pulse, Λ = v F /a for the high-energy cutoff of the chiral Luttinger liquid theory and P l = |p l | 2 for the probability for a quasiparticle to absorb (l > 0) or emit (l < 0) l photons [10]. At finite temperature, ∆S needs to be slightly amended,

∆S = S -2e * I B (t) coth q 2θ = - 2 T e * Γ 0 v F 2 2θ Γ(2ν) 2πΩθ Λ 2ν-2 × l P l Γ ν + i l + q 2πθ 2 sinh l 2θ sinh q 2θ , (B14)
in order to get rid of the thermal noise (∆S → 0 for large temperature). There, the reduced temperature is θ = k B Θ Ω (Θ the electron temperature) and the finitetemperature expression of G (-τ ) has been employed. ) This form of H T (specific to ν = 1/2) allows us to refermionize the e iφ-field, and to decouple φ + , following Ref. [42]. A new fermionic field ψ(x) and a Majorana fermion field f , which satisfy {f, f } = 2 and {f, ψ(x)} = 0, are introduced. These obey the equations of motion:

H 0 = v F 4π r=± dx (∂ x φ r ) 2 , (D1)
H T = Γ(t) e iφ-(0) 2πa + Γ * (t) e -iφ-(0) 2πa . ( D2
-i∂ t ψ(x, t) =iv F ∂ x ψ(x, t) + Γ(t) √ 2πa f (t)δ(x), (D3) -i∂ t f (t) =2 1 √ 2πa [Γ * (t)ψ(0, t) -H.c.] . (D4)
Solving this set of equations near the position x = 0 of the quantum point contact (QPC), one can relate the fields ψ b and ψ a corresponding to the new fermionic field taken respectively before and after the QPC:

ψ a (t) =ψ b (t) -γΩe iϕ(t)+iqΩt t -∞ dt e -γΩ(t-t ) × e -iϕ(t )-iqΩt ψ b (t ) -H.c. . (D5)
The backscattered current is the difference of the leftmoving current after and before the QPC:

I B (t) = ev F 2 ψ † b (t)ψ b (t) -ψ † a (t)ψ a (t) . ( D6
)
After some algebra, the time-averaged backscattered current becomes:

I B = - e T γ l P l Im Ψ 1 2 + γ -i(q + l) 2πθ , (D7)
where Ψ(z) is the digamma function, and the dimensionless tunneling parameter is γ = |Γ0| 2 πav F Ω . Similarly, the zero-frequency time-averaged shot noise defined in Eq. (B11) takes the form: Finally, the excess shot noise is obtained using the known θ = 0 DC results [42] for the backscattered current I B dc and corresponding zero-frequency noise S dc : The excess noise at θ = 0 associated with an arbitrary drive V (t) is then defined as the difference between the photo-assisted shot noise (PASN) and the DC noise S dc (Q T ) obtained for the same charge Q T = T I B transferred during one period of the AC drive ∆S = S -2e * I B + (e * )

I B dc = e 2π ξ arctan eV dc 2ξ , (
In the present work, we apply the voltage to a long contact, so that V 1 (x, t) = θ(-x -d)V (t). This leads to a phase shift of the form

Φ 1 = t -∞ dt V 1 (x -v F (t -t ), t ) = t -∞ dt θ(-x + v F (t -t ) -d)V (t ) = t-x+d v F -∞ dt V (t ).
(F2)

Following now Ref. 9, one can recast the single particle Hamiltonian defined in their Eq. ( 4) into a form similar to the one presented here in Eq. (B3) provided that one defines the applied voltage drive as V 2 (x, t) = v F δ(x + d) t -∞ dτ V (τ ). This, in turn, leads to the phase shift

Φ 2 = t -∞ dt V 2 (x -v F (t -t ), t ) = t -∞ dt v F δ(x -v F (t -t ) + d) t -∞ dτ V (τ ) = t-x+d v F -∞ dτ V (τ ) (F3)
One thus readily sees that at the level of the phase shift experienced by the quasiparticles as a result of the external drive, the protocol presented in Ref. 9 and the one presented in the text are completely equivalent.

Figure 1 .

 1 Figure 1. Main setup: a quantum Hall bar equipped with a QPC connecting the chiral edge states of the FQH. The left-moving incoming edge is grounded at contact 2 while the right-moving one is biased at contact 1 with a time-dependent potential V (t).

Figure 2 .

 2 Figure 2. Excess noise in units of S 0 = 2

Figure 3 .

 3 Figure 3. Rescaled excess noise ∆S/γ in units of e 2T as a function of the number of electrons per pulse q, for different values of the dimensionless tunneling parameter γ = |Γ 0 | 2 πav F Ω . Results are obtained at zero temperature, with filling factor ν = 1/2, in the case of a square (bottom), a cosine (middle) and a periodic Lorentzian drive with η = 0.1 (top).

Figure 4 .

 4 Figure 4. Normalized HOM noise ∆Q at q = 1, as a function of the time delay τ between pulses. Results are presented in the WB case at ν = 1/3 and θ = 0.1, for a square, a cosine and a periodic Lorentzian drive with η = 0.1. Inset: HOM setup with applied drives on both incoming arms.

Figure 5 .

 5 Figure 5. Averaged backscattered current IB(t) as a function of the number of electrons per pulse q, in the case of a square, cosine and periodic Lorentzian drive with η = 0.1 and θ = 0.1. Results are presented for ν = 1/3 in the perturbative regime in units of I 0 = e * T Γ 0 v F 2 Ω Λ 2ν-2 (top) and for the exact treatment at ν = 1/2 in units of e T (bottom).

S = e 2 T 4γ 2 klmp * k p l p * l+m p k+m m 2

 22 Re

  where we introduced ξ = |Γ0| 2 πav F . This allows to express the DC noise, not as a function of the applied bias, but rather as a function of the charge Q ∆t = ∆t I B dc transferred through the QPC over a given time interval ∆tS dc (Q ∆t ) = e Q ∆t ∆t -

ACKNOWLEDGMENTS

We are grateful to G. Fève, B. Plaçais, P. Degiovanni and D.C. Glattli for useful discussions. This work was granted access to the HPC resources of Aix-Marseille Université financed by the project Equip@Meso (Grant No. ANR-10-EQPX-29-01). It has been carried out in the framework of project "1shot reloaded" (Grant No. ANR-14-CE32-0017) and benefited from the support of the Labex ARCHIMEDE (Grant No. ANR-11-LABX-0033) and of the AMIDEX project (Grant No. ANR-11-IDEX-0001-02), all funded by the "investissements d'avenir" French Government program managed by the French National Research Agency (ANR).

Appendix A: External drives and corresponding Floquet coefficients

The applied drive V (t) is split into a DC and an AC part V (t) = V dc + V ac (t), where by definition V ac (t) averages to zero over one drive period T .

The AC voltage is handled through the accumulated phase experienced by the quasiparticles ϕ(t) = e * t -∞ dt V ac (t ) (with the fractional charge e * = νe). We use the Fourier decomposition of e -iϕ(t) , defining the corresponding coefficients p l as

e ilΩt e -iϕ (t) .

(A1)

We focus on three types of drives:

where rect(x) = 1 for |x| < 1/2 (= 0 otherwise), is the rectangular function, and η = W/T (W is the half-width at half-maximum of the Lorentzian pulse).

The corresponding Fourier coefficients of Eq. (A1) read, for non-integer q = e * V dc Ω :

Cosine p l = J l (-q) , (A5)

Lorentzian

Appendix B: Current and noise in the weak backscattering regime

Fractional quantum Hall (FQH) edges at filling factor ν = 1/(2n + 1) are described in terms of a hydrodynamical model [37] through the Hamiltonian of the form

where we apply a bias V (x, t) which couples to the charge density of the right moving edge state. Here the bosonic fields satisfy [φ R,L (x), φ R,L (y)] = ±iπSgn(x -y). These bosonic fields propagate along the edge at velocity v F and are directly related to the corresponding The number of excitations created in a onedimensional system of free fermions by the applied timedependent drive V (t) is given by the number of electrons and holes

where n F (k) is the Fermi distribution and ψ k is a fermionic annihilation operator in momentum space. Using the bosonized description for ν = 1 [see Eq. (B4)], the number of electrons and holes becomes

Minimal excitations correpond to a drive which excites a single electron, while no particle-hole pairs are generated (N h = 0). Generalizing this to a chiral Luttinger liquid (a FQH edge state) [37], this excitation should correspond to a vanishingly small value of the quantity

The excess noise [defined in Eq. (B12)] is thus identified as the most suitable quantity to study minimal excitations. At θ = 0, one recovers in N precisely the excess noise, up to a prefactor which depends on the tunneling amplitude [see Eq. (B12)] [9]. At θ = 0, thermal excitations contribute substantially to N , motivating us to include a correction [the coth factor in Eq. (B14)] which gets rid of this spurious contribution in the high temperature limit. A periodic voltage bias is now applied to both rightand left-moving incoming arms of the QPC. We focus here on single leviton collisions with identical potential drives, up to a tunable time delay τ . The drives are periodic Lorentzians with a single electron charge per pulse (q R = q L = 1). Using a gauge transformation, this amounts to computing the noise in the case of a single total drive V Tot (t) = V (t) -V (t -τ ) applied to the right incoming branch only.

In the context of electron quantum optics, a standard procedure is to compare this so-called Hong-Ou-Mandel (HOM) noise to the Hanbury-Brown and Twiss (HBT) case where single levitons scatter on the QPC without interfering, leading to the definition of the following normalized HOM noise [5,24]

Thermal fluctuations are eliminated by subtracting the vacuum contribution S vac to each instance of the noise. Taking advantage of the gauge transformation, we use the expressions for the noise established earlier in the perturbative and the exact cases. This calls for a new set of Fourier coefficients pl associated with the total drive

where z = e -2πη and the Fourier coefficients p l corresponding to a periodic Lorentzian drive V (t) with q = 1 take the form

with θ H (x) the Heaviside step function.

In the WB case, the PASN is given by Eq. (B11), which gives at finite temperature

while the vacuum contribution reduces to

Combining the results from Eqs. (E1) through (E6), we obtain for the normalized HOM noise

Remarkably, this result is independent of both temperature and filling factor. Indeed, thermal contributions factorize in the exact same way in the numerator and denominator, leading to a universal profile. This result also corresponds to that of Ref. [4] for ν = 1.

Appendix F: Applying the voltage bias to a point-like or a long contact

In the main text, we focus on the experimentally relevant case of a long contact [4] where electrons travel a long way through ohmic contacts before reaching the mesoscopic conductor, accumulating a phase shift along the way. In a previous work [9] however, the authors consider applying the voltage pulse through a point-like contact. Here we show that these two approaches are equivalent.

Indeed, one can see starting from the Hamiltonian (B1), and solving the corresponding set of equations of motion for the fields that the external bias can be accounted for by implementing a phase shift of the quasiparticle operator, which we recall here ψ R (x, t) -→ ψ R (x, t) e -iνe t -∞ dt V (x-v F (t-t ),t ) . (F1)

Several choices for the external drive V (x, t) are thus acceptable, provided that the integral t -∞ dt V (xv F (t -t ), t ) leads to the same result. Indeed, this phase shift is the only meaningful physical quantity, which gives us some freedom in the choice of V (x, t). We further consider two different options.


Majda Mekić

Brice Temime-Roussel

Anne Monod

Rafal S Strekowski
email: rafal.strekowski@univ-amu.fr

Quantification of gas phase methyl iodide using H 3 O + as the reagent ion in the PTR-MS technique

Keywords: Methyl iodide, CH 3 I, H 3 O +, PTR-MS, collision rate

In this work, the proton transfer reaction between the hydronium ion (H 3 O + ) and methyl iodide (CH 3 I) is studied to investigate if consistent quantification of the gas phase CH 3 I is possible in humid air. The neutral CH 3 I molecule was chosen because this compound is of environmental importance in the field of nuclear power plant safety and nuclear energy. Water was used as a reagent ion source in a conventional Insbruck PTR-MS to produce H 3 O + reagent ions. The use of H 3 O + ions allows for fast, sensitive and specific detection of gas phase CH 3 I via a proton-transfer reaction

The instrument response was linear in the tested 5 to 96 ppbV range and the PTR-MS sensitivity was observed to be humidity dependent. The observed sensitivity was in found to range between 1.6 to 3.3 cps/ppb at relative humidity between 63 and 15% at T= 23 • C. A typical H 3 O + primary ion signal was 10 7 cps and the normalized sensitivity was in the range between 0.16 and 0.33 ncps/ppb. The instrument CH 3 IH + ion background rate was 6.8 ± 1.4 cps and the dwell time was 1 second. The detection limit was calculated as 3 times the standard deviation of the background level and ranged between 1.3 and 3.8 ppb.

The theoretical collision rate based on the dipole moment and molecular polarizability is calculated. The theoretical collision rate is compared with the experimentally obtained values. The results indicate that the PTR-MS technique is a good analytical method to detect and quantify gas phase CH 3 I concentrations.

Introduction

The atmospheric importance of molecular iodine and alkyl iodides was first suggested by Chameides and Davis (1982). [START_REF] Chameides | Iodine: Its possible role in tropospheric photochemistry[END_REF] It is now well understood that iodine and iodine containing volatile organic compounds play an important role in the oxidizing capacity of the troposphere [START_REF] Bloss | Impact of halogen monoxide chemistry upon boundary layer oh and ho 2 concentrations at a coastal site[END_REF], aerosol formation and in the ozone depleting cycles in the troposphere [START_REF] Davis | Potential impact of iodine on tropospheric levels of ozone[END_REF][START_REF] Mcfiggans | A modeling study of iodine chemistry in the marine boundary layer[END_REF] and stratosphere. [START_REF] Solomon | On the role of iodine in ozone depletion[END_REF] In addition to atmospheric interest, the presence of organic iodides has gained an increased interest in the field of nuclear industry safety to better understand chemical processes responsible for the formation of different fission products if a major nuclear power plant accident type Three Mile Island (U.S.A.) [START_REF] Cline | Measurements of 129 i and radioactive particulate concentrations in the tmi-2 containment atmosphere during and after the venting[END_REF], Chernobyl (Ukraine) [START_REF] Noguchi | Physicochemical speciation of airborne 131i in japan from chernobyl[END_REF] and Fukushima (Japan) [START_REF] Kinoshita | Assessment of individual radionuclide distributions from the fukushima nuclear accident covering central-east japan[END_REF] were to occur, again. It is now known that iodine and methyl iodide are two of the more critical fission products that are released from UO 2 feul during a major nuclear power plant accident in light water form of the fission product iodine (I 2 ) that is difficult to retain by post-accident filtration systems. [START_REF] Ball | Behaviour of iodine project: Final report on organic iodide studies[END_REF] Further, methyl iodide is currently used in the nuclear industry field to test the organic iodine capture ability and ageing on the performance of emergency charcoal filters. [START_REF] Nacapricha | Quality control of nuclear charcoals: Particle size effect and trapping mechanism[END_REF][START_REF] Wren | Methyl iodide trapping efficiency of aged charcoal samples from bruce-a emergency filtered air discharge systems[END_REF] Methyl iodide has additional applications in the agriculture industry where it has been introduced as a fumigant pesticide used to control insects, plant parasitic metabolites, soil-bome pathogens and weed seeds. [START_REF] Waggoner | Methyl iodide: an alternative to methyl bromide for insectary fumigation[END_REF] To date, most environmental CH 3 I measurements are based on gas chromatographic (GC) separation equipped with a mass spectrometer (MS) or an electron capture detector (ECD). A more recent atmospheric CH 3 I measurement method is based on resonant fluorescence (RF) spectroscopy. [START_REF] Bale | Novel measurements of atmospheric iodine species by resonance fluorescence[END_REF] While GC-ECD (LD∼10 ppt) and GC-MS (LD∼100 ppt) methods are two very sensitive analytical techniques used to detect gas phase CH 3 I in the laboratory and field settings, they do not offer the time resolution and response needed to measure rapid flux or concentration changes in the gas phase. The newer RF technique used to detect gas phase CH 3 I in the laboratory and more recently in the field is fast, sensitive and selective but remains a research-grade instrument that requires advanced technical expertise and skills. The chemical ionization mass spectrometric (CIMS) technique has the potential for fast, sensitive, specific and real time CH 3 I measurements when rapid changes in the gas phase mixing ratios need to be known or monitored continuously.

The proton-transfer-reaction mass spectrometry (PTR-MS) is a type of CIMS instrument. The PTR-MS combines the concept of chemical ionization [START_REF] Field | Chemical ionization mass spectrometry. i. general information[END_REF] with the flow-drift-tube technique. [START_REF] Mcfarland | Flow-drift technique for ion mobility and ion-molecule reaction rate constant measurements. ii. positive ion reactions of n + , o + , and h + 2 with o 2 and o + with n 2 from thermal to [inverted lazy s] 2 ev[END_REF] While protontransfer-reaction mass spectrometry (PTR-MS) is often used for sensitive detection of volatile organic compounds (VOCs), it can be applied only to gas-phase compounds with proton affinities (PA) higher than that of water, PA(H 2 O) = 691.0 ± 3 kJ mol -1 [START_REF] Hunter | Evaluated gas phase basicities and proton affinities of molecules: an update[END_REF][START_REF]Webbook de chimie NIST[END_REF], and the proton transfer from H 3 O + to the analyte molecule is efficient if the difference in proton affinities is larger than ∼ 35 kJ mol -1 . [START_REF] Bouchoux | A relationship between the kinetics and thermochemistry of proton transfer reactions in the gas phase[END_REF] The proton affinity of CH 3 I, PA(CH 3 I) = 691.7 kJ mol -1 , [START_REF] Hunter | Evaluated gas phase basicities and proton affinities of molecules: an update[END_REF][START_REF]Webbook de chimie NIST[END_REF] is only slightly higher than that of water.

Since the difference in proton affinities between water and methyl iodide is very small, the CIMS technique based on proton transfer reaction of hydronium (H 3 O + ) reagent ions with CH 3 I has only been used once [START_REF] Spanel | Selected ion flow tube studies of the reactions of h3o+, no+, and o2+ with several aromatic and aliphatic monosubstituted halocarbons[END_REF] in a laboratory setting to detect methyl iodide. Further, the one previous work has been carried out in dry helium flow only. In this work it is shown that H 3 O + ions may still be employed to detect gas phase CH 3 I. The use of water (H 2 O) as a source of reagent ions in the PTR-MS instrument to detect CH 3 I is proposed. H 3 O + reagent ions are used for sensitive and specific detection of gas phase CH 3 I in humidified and dry air. H 3 O + ion is shown to be a good proton source and soft chemical ionization reagent. The proposed PTR-MS technique appears to be a good tool for online analyses of relatively fast changing concentrations of methyl iodide.

Experimental

The proton transfer reaction between the H 3 O + ions and CH 3 I was studied using a commercial quadrupole PTR-MS mass analyzer (Ionicon Analytik GmbH, Innsbruck, Austria). The experimental details that are particularly relevant to this work are given below.

Generation of gas phase CH 3 I. The gas phase CH 3 I concentration was generated using the gas saturation method, one of the oldest and most versatile ways of studying heterogeneous equilibria involving low vapor pressure compounds, first developed by Regnault in 1845. [START_REF] Regnault | études sur l'hygrométrie[END_REF] The gas saturation method used in this work is similar to the one described originally by [START_REF] Markham | A compact gas saturator[END_REF]. [START_REF] Markham | A compact gas saturator[END_REF] Briefly, nitrogen carrier gas was allowed to flow though the volume containing the CH 3 I sample that itself was mixed with glass beads and supported on a fritted glass surface. The saturator volume itself was immersed in a temperature controlled fluid and kept at a constant temperature using a thermostat with an accuracy of ±0.1K.

The temperature inside the saturator volume was measured using a Type-J thermocouple (Omega) with an accuracy of ±0.1K. The carrier gas was allowed to enter the saturator volume, equilibrate with the sample and was then allowed to exit through a capillary passageway and allow to flow through a glass tube. The geometry of the exiting glass tube was such that the diameter of the glass tube increased with increasing length. This was done to avoid any sample condensation as the sample and the carrier gas were allowed to leave the saturator system. Concentration of CH 3 I at the exit of the saturation system was calculated from the given vapor pressure, mass flow rates, pressure within the saturator and the total pressure. The Antoine type equation used to calculate the vapor pressure of CH 3 I is log p = -20.3718 -1253.6/T + 13.645 log T -2.6955 × 10 -2 T + 1.6389 × 10 -5 T 2 where the pressure p is in units of mmHg and the temperature T is in Kelvin and in the range from 207.7K to 528.00K. [START_REF] Yaws | Chemical properties handbook : physical, thermodynamic, environmental, transport, safety, and health related properties for organic and inorganic chemicals[END_REF].

Under normal operating conditions, the saturator was kept at T = 268 K. At this temperature, the vapor pressure of CH 3 I within the saturator was calculated to be 109.31 mmHg. This vapor pressure was then further diluted using nitrogen gas carrier gas and a system of mass flow controllers to obtain the desired concentration.

PTR-MS instrument.

A commercial PTR-MS (Ionicon Analytic GmbH, Innsbruck, Austria) was used to study the H 3 O + reagent ion ionization process with CH 3 I. [START_REF] Hansel | Proton transfer reaction mass spectrometry: on-line trace gas analysis at the ppb level[END_REF][START_REF] Lindinger | On-line monitoring of volatile organic compounds at pptv levels by means of proton-transfer-reaction mass spectrometry (ptr-ms) medical applications, food control and environmental research[END_REF] The reaction chamber pressure (p drift ) was 2.11 mbar, drift tube voltage was 601V and the drift tube temperature was approximately 310K, albeit it was not controlled. The corresponding E/N ratio was 134 Td (1Td=10 -17 Vcm 2 ) where E is the electric field strength (E) applied to the drift tube and N is the buffer gas density. This E/N ratio value was chosen to limit clustering of the H 3 O + reagent ions with H 2 O because it is known that the resulting cluster ions (m/Q = 37.0501 and m/Q = 55.0395) may act as reagent ions, and, as a result, limit signal intensity. [START_REF] Hansel | Proton transfer reaction mass spectrometry: on-line trace gas analysis at the ppb level[END_REF][START_REF] Lindinger | On-line monitoring of volatile organic compounds at pptv levels by means of proton-transfer-reaction mass spectrometry (ptr-ms) medical applications, food control and environmental research[END_REF] The effect of the water cluster ion formation on signal intensity [START_REF] De Gouw | Validation of proton transfer reaction-mass spectrometry (ptr-ms) measurements of gas-phase organic compounds in the atmosphere during the new england air quality study (neaqs) in 2002[END_REF] was assessed by changing the water content within the carrier gas. Here, nitrogen gas was allowed to pass through a bubbler filled with deionized water at room temperature. The resulting relative humidity (%RH) was measured at the inlet of the PTR-MS using a temperature-humidity mini probe (HygroClip-SC04, Rotronic International). The relative humidity was varied between 0 and 60%. Data analysis. PTR-MS data files were imported using the Technical Data Management (TDM) Excel add-in for Microsoft Excel. Mass spectra were recorded up to m/Q 200 and the integration rate was 1s.

No specific mass calibration was performed. However, stability of the hydronium ion signal at m/Q 19 was checked. The Excel raw data files were exported and analyzed using Igor Pro 6.37 commercial software.

Materials. The nitrogen carrier gase used in this study was generated using the N2LCMS 1 Nitrogen Generator (Claind S.r.l., Italy). Iodomethane was purchased at Acros Organics (Belgium) and the stated minimum purity was 99%. To limit any photo-catalytic or thermal decomposition during storage, CH 3 I original container bottles were stored under dark conditions at T = 6 • C. Deionized water with a resistivity greater than 18M was prepared by allowing tap water to pass first through a reverse osmosis demineralization filter (ATS Groupe Osmose) and then through a commercial deionizer (Milli-pore, Milli-Q).

Results

Experimental determination of k (H3O + +CH3I)

Based on the original work of Lindinger and coworkers (1998) [START_REF] Lindinger | On-line monitoring of volatile organic compounds at pptv levels by means of proton-transfer-reaction mass spectrometry (ptr-ms) medical applications, food control and environmental research[END_REF], the CH 3 I gas phase concentration may be calculated using the following equation.

[

CH 3 I] cm -3 = 1 kt • CH 3 I -H + H 3 O + • Tr H3O + Tr CH3I-H + (1)
In the equation ( 1) above, CH 3 I-H + and H 3 O + are the ion count rates, k is the rate coefficient of the proton transfer reaction of the H 3 O + reagent ion with CH 3 I and t is the residence or reaction time of the ion within the drift tube (∼ 100 µs). [START_REF] Cappellin | Proton transfer reaction rate coefficients between h 3 o + and some sulphur compounds[END_REF] The reaction time t is calculated using the length of the drift tube l, ion mobility µ, electric field E and the electric potential applied to the drift tube U drift . Given that

l t = µ • E = µ • U drift l (2)
therefore,

t = l 2 µ • U drift (3)
The length of the drift tube l = 9.3 cm and U drift is in units of Volts. The ion mobility µ listed in equations 2 and 3 is calculated using the following equation 4

µ = µ 0 • p 0 p drift • T drift T 0 (4)
where the reduced mobility µ 0 = 2.8 cm 2 V -1 s -1 , pressure p 0 = 1013.25 mbar, and temperature at standard conditions T 0 = 273.15 K. As a result, the reaction time within the drift-tube may be calculated using the following equation ( 5)

t = l 2 µ 0 • U drift • T 0 T drift • p 0 p drift (5)
where T drift is the drift-tube temperature in Kelvin and p drift is the drift-tube pressure in millibar. Using the ideal gas law, the number of air molecules per cm 3 within the drift-tube volume may be calculated using the following equation 6

air cm -3 = N A 22400 • T 0 T drift • p 0 p drift (6)
where the Avogadro's number N A = 6.022 × 10 23 mole -1 . As a result, the mixing ration of CH 3 I in the gas phase detected by the PTR-MS may be determined using the following equation.

[

CH 3 I] ppbV = 10 9 k • 22400 • µ 0 • U drift N A • l 2 • T 2 drift T 2 0 • p 2 0 p 2 drift • CH 3 I -H + H 3 O + • Tr H3O + Tr CH3I-H + (7)
The equation ( 7) may be simplified by including the constant factors listed above to give the following equation 8.

[ If the rate coefficient for the proton transfer reaction of the H 3 O + ion with CH 3 I is known, the gas phase mixing ratio of methyl iodide may be determined using the PTR-MS technique without prior calibration by simply using MS signal ion counts and given operating instrument parameters. To the best of our knowledge, there is no rate coefficient listed in the literature for the proton transfer reaction between H 3 O + ion and CH 3 I. As a result, the k value has been determined experimentally and theoretically. The equation [START_REF] Kinoshita | Assessment of individual radionuclide distributions from the fukushima nuclear accident covering central-east japan[END_REF] above may be rewritten in the following form to give k exp , that is, the experimental rate coefficient for the proton transfer H 3 O + + CH 3 I reaction.

CH 3 I] ppbV = 1.657 × 10 -11 • U drift • T 2 drift k • p 2 drift • CH 3 I -H + H 3 O + • Tr H3O + Tr CH3I-H + (8)
k exp = 1.657 × 10 -11 • U drift • T 2 drift [CH 3 I] ppbV • p 2 drift • CH 3 I -H + H 3 O + • Tr H3O + Tr CH3I-H + (9)
Equation ( 9) may then be used to determine k if CH 3 I mixing ratio is known.

A typical signal intensity profile of selected ions is shown in Figure 1. As shown in Figure 1, m/Q 21 refers to the hydronium ion H 18 3 O + , m/Q 37 refers to the water complex ion H 3 O + (H 2 O) (other water clusters have been ignored [START_REF] Schwarz | Determining concentration patterns of volatile compounds in exhaled breath by ptr-ms[END_REF]), m/Q 143 refers to the CH 3 I-H + ion and m/Q 142 refers to the CH 3 I + ion. The primary ions H 3 O + at m/Q 19 were not detected but were calculated based on the ion count rates of H 18 3 O + at m/Q 21 using equation 10.

count rate m/Q 19 = (count rate m/Q 21) × 500 Transmission H 18 3 O + ion (10)
In equation 10 above, the constant 500 is the isotope ratio that reflects the isotope ratio of H 18 3 O + to H 16 3 O + and the intensity of the H 18 3 O + ion is corrected by its transmission efficiency. Product ions of reactions caused by minor impurities, namely NO + and O + 2 , have been shown to have negligible intensities as compared to all other product ions and were not considered. [START_REF] Schwarz | Determining concentration patterns of volatile compounds in exhaled breath by ptr-ms[END_REF] The ion signal intensity at m/Q 143 shown in Figure 1 is used to relate the PTR-MS response to methyl iodide mixing ratio. A typical calibration plot is shown in Figure 2. As shown in Figure 2, the MS response is linear for the given methyl iodide mixing ratios. Under the typical experimental conditions used, no fragmentation of the product ion was observed. That is, only the mother ion at m/Q 143 that refers to the CH 3 I-H + ion was detected. The difference in proton affinities between water and methyl iodide is very small. As a result, there is no sufficient excess of energy to fragment the product ion during the proton transfer ionization process. However, as shown in Figure 1 the signal intensity of the m/Q 142 that refers to the CH 3 I + ion was observed to increase slightly with increasing methyl iodide mixing ratio. It is not believed that the m/Q 142 is a result of the fragmentation of the methyl iodide mother ion. It is known that a small amount of the O + 2 ion is always present within the drift tube (see Figure 1). The O + 2 reagent ion will react with methyl iodide via an electron transfer to produce m/Q 142 ion signal that refers to the CH 3 I + ion.

Comparison of reported measured rate coefficients for the proton transfer reaction between H 3 O + and CH 3 I at different humidities is shown in Table 1. The methyl iodide mixing ratio, [CH 3 I] saturator , shown in Table 1 was determined at the exit of the saturation system and calculated from the given temperature, vapor pressure, mass flow rates, pressure within the saturator and total pressure.

Theoretical determination of k (H3O + +CH3I)

The theoretical value of the collision rate constant for the proton transfer reaction between H 3 O + and CH 3 I was calculated using the average dipole orientation (ADO) theory. [START_REF] Su | Ion-polar molecule collisions: the effect of ion size on ion-polar molecule rate constants; the parameterization of the average-dipole-orientation theory[END_REF][START_REF] Su | Theory of ion-polar molecule collisions. comparison with experimental charge transfer reactions of rare gas ions to geometric isomers of difluorobenzene and dichloroethylene[END_REF] The ADO rate coefficient, k ADO , is given by

k ADO = q πα µε 0 + C qµ D ε 0 1 2πµk B T (11)
where µ D and α are the dipole moment and polarizability of the neutral molecule, respectively, q is the fundamental charge, ε 0 is the permittivity of the free space, µ is the reduced mass of the colliding partners

d k ADO = 2.28 × 10 -9 cm 3 s -1
(here, H 3 O + and CH 3 I), T is the temperature within the drift-tube and k B is the Boltzmann constant.

The dipole locking constant C is between 0 and 1 and is a function of µ D / √ α and temperature. [START_REF] Su | Parameterization of the average dipole orientation theory: temperature dependence[END_REF] The structure of methyl iodide was taken from NIST database. [START_REF]Webbook de chimie NIST[END_REF] In this work, the dipole moment µ D = 1.620 Debye and polarizability α = 7.325 Å3 . [START_REF]Webbook de chimie NIST[END_REF] The theoretical rate constant for the protontransfer reaction between H 3 O + and CH 3 I is calculated to be k ADO = 2.28 × 10 -9 cm 3 s -1 .

Discussion

MS ion signal

It was observed that the ion signal sensitivity decreased as a function of relative humidity. The relationship between sensitivity and relative humidity is shown in Figure 3. As shown in Figure 3, the CH 3 IH + ion signal was normalized to the signal of H 3 O + ion (m/Q 19) because the reagent ion also changes with humidity. Since the CH 3 IH + ion signal is proportional to the H 3 O + ion abundance, it needs to be taken into account to show the humidity dependence of the CH 3 IH + ion. The observed drop is sensitivity with increasing relative humidity is best explained by the water cluster formation at low drift-tube E/N ratios. For example, the plot of the fraction of the dihydrate cluster in the drift tube as a function of relative humidity and drift-tube E/N value is shown in Figure 4. It can be seen in Figure 4 that a low drift-tube E/N value promotes clustering of H 3 O + ions with H 2 O. The formation of water clusters may be problematic since they may act as reagent ions. To limit cluster formation within the drift-tube, all experiments were performed at E/N = 140 Td. In a similar study, Cappellin et al. ( 2012) [START_REF] Cappellin | On quantitative determination of volatile organic compound concentrations using proton transfer reaction time-of-flight mass spectrometry[END_REF] reported that under high electric field strength values, the protonated water cluster ion formation is strongly suppressed. Further, since the proton transfer reaction between the H 3 O + ion and CH 3 I is only slightly exothermic, and possibly endothermic given the uncertainties in the proton affinities, the proton transfer reaction between the water cluster and CH 3 I will most likely not occur. For example, the water dimer (H 2 O) 2 , has a proton affinity of 808 ± 6kJ mol -1 . As a result, the increased affinity of the water cluster means that some reactions that occur with H 3 O + ions will not take place with H 3 O + (H 2 O) ions.

Rate constants

The agreement between the theoretical value for the rate constant for the proton-transfer reaction between H 3 O + and CH 3 I (k ADO ) and the experimental value (k exp ) is very bad (see Table 1). This is not surprising given that the proton-transfer reaction H 3 O + + CH 3 I is only slightly exothermic (∆PA = 0.7 kJ mol -1 ). Further, given the uncertainties in proton affinities (±3 kJ mol -1 ), the reaction itself may be endothermic. It has been argued that the proton-transfer from H 3 O + to the analyte molecule in the drift-tube of the PTR-MS is efficient only when the molecule has a higher proton affinity and that the difference in proton affinities is larger than 35 kJ mol -1 . [START_REF] Bouchoux | A relationship between the kinetics and thermochemistry of proton transfer reactions in the gas phase[END_REF] The experimental values for the proton transfer reaction between H ratio, where [CH 3 I] PTR-MS is the theoretically predicted CH 3 I mixing ratio estimated by the PTR-MS measurements (assuming k ADO = 2.28 × 10 -9 cm 3 s -1 [START_REF] Lindinger | Proton-transfer-reaction mass spectrometry (ptr-ms): on-line monitoring of volatile organic compounds at pptv levels[END_REF]) and [CH 3 I] saturator is the CH 3 I mixing ratio determined at the exit of the saturation system and calculated from the given vapor pressure, mass flow rates, pressure within the saturator and total pressure. The possible explanation for the low k exp /k ADO ration is a reverse reaction of the product CH 3 I-H + ion with the molecules of H 2 O even under low humidity conditions present within the drift-tube. This hypothesis is supported by the data shown in Figure 3.

The proton transfer reaction between H 3 O + and CH 3 I has been studied briefly by [START_REF] Spanel | Selected ion flow tube studies of the reactions of h3o+, no+, and o2+ with several aromatic and aliphatic monosubstituted halocarbons[END_REF]. [START_REF] Spanel | Selected ion flow tube studies of the reactions of h3o+, no+, and o2+ with several aromatic and aliphatic monosubstituted halocarbons[END_REF] These authors calculated the k exp /k collision = 0.5, where k collision is the collision rate constant in dry helium carrier gas calculated based on the work of [START_REF] Su | Parametrization of the ion-polar molecule collision rate constant by trajectory calculations[END_REF]. [START_REF] Su | Parametrization of the ion-polar molecule collision rate constant by trajectory calculations[END_REF] In the work presented here, the k exp /k ADO = 0.02 in dry synthetic air. At this point, we cannot determine the discrepancy between the two values. Since [START_REF] Spanel | Selected ion flow tube studies of the reactions of h3o+, no+, and o2+ with several aromatic and aliphatic monosubstituted halocarbons[END_REF] In a scenario similar to this work, Cappellin and coworkers (2014) [START_REF] Cappellin | Ethylene: Absolute real-time high-sensitivity detection with ptr/sri-ms. the example of fruits, leaves and bacteria[END_REF] determined the rate coefficient for the reaction of ethylene with H 3 O + . Similar to the results obtained in this work, Cappellin and coworkers' theoretical value k ADO for the ethylene + H 3 O + reaction does not agree with the k exp value. These authors reported the experimental reaction rate coefficient between 1.7×10 -11 cm 3 s -1 and 3.4×10 -11 cm 3 s -1 , far below the collision rate (1.4 × 10 -9 cm 3 s -1 ). [START_REF] Cappellin | Ethylene: Absolute real-time high-sensitivity detection with ptr/sri-ms. the example of fruits, leaves and bacteria[END_REF] Similar to the results observed in this work, the reaction of ethylene with H 3 O + is very inefficient. The authors attribute the observed inefficiency to the (quasi) reaction endothermicity. The results obtained in this work for the proton-transfer reaction of H 3 O + with methyl iodide are consistent with the results obtained by Cappellin and coworkers [START_REF] Cappellin | Ethylene: Absolute real-time high-sensitivity detection with ptr/sri-ms. the example of fruits, leaves and bacteria[END_REF].

In this work, the observed difference in k exp and k ADO for the H 3 O + + CH 3 I reaction does not imply that the PTR-MS technique is a poor method of choice to be used to detect gas phase CH 3 I. As shown in Figures 1 and2, with a good calibration curve, the PTR-MS technique is a perfect analytical method to detect and quantify rapid changes in the CH 3 I gas phase concentrations. the rate constant is assumed to be the result of the fact that the proton transfer reaction between the hydronium ions and methyl iodide is only slightly exothermic and given the uncertainties in the proton affinities may be even endothermic.

Conclusion

Figure 1 :

 1 Figure 1: Typical signal ion counts (Hz) of selected ions.

Figure 2 : 7 (

 27 Figure 2: Plot of the methyl iodide calibration results. The m/Q CH 3 I signal has been normalized by m/Q 19. The solid line is obtained from a linear least squares analysis; its slope is (2.21 ± 0.04) × 10 -7 and the intercept is (3.85 ± 2.5) × 10 -7(Errors are 2σ, precision only).

Figure 3 :

 3 Figure 3: Plot of the signal ion sensitivity as a function of relative humidity.

Figure 4 :

 4 Figure 4: Plot of the fraction of the dihydrate cluster in the drift tube versus relative humidity. Experimental conditions: T = 298K, E/N = 140Td (open circles •), E/N = 127Td (closed circles •). The percentage data of dihydrate clusters were calculated using the following equation: [H 2 O-H 18 3 O + ]×250 [H 18 3 O + ]×500× 100

A

  commercial Ionicon Analytic PTR-MS was used to study the H 3 O + reagent ion ionization process with methyl iodide. The proton transfer reaction between the H 3 O + reagent ion and CH 3 I produces a specific ion at m/Q 142 (CH 3 I-H + ) that allows for fast and sensitive detection of gas phase methyl iodide. The instrument response for CH 3 I was shown to be linear in the 5 -96 ppbV range. The calculated collision rate, k ADO is calculated based on a dipole moment µ(CH 3 I) = 1.620 D [18] and a molecular polarizability α(CH 3 I) = 7.325 Å3 [18] and gives k ADO = 2.28 × 10 -9 cm 3 s -1 . The experimental collision rate, k exp for the proton transfer reaction H 3 O + + CH 3 I → [CH 3 I-H] + + H 2 O is in the range (1.79 -4.90) × 10 -11 cm 3 s -1 . The observed difference in the theoretical and experimental values of

Table 1 :

 1 Comparison of the measured rate coefficients for the proton transfer reaction between H 3 O + and CH 3 I at different

	humidities
	[CH 3 I] saturator range a	%RH b	k a,c exp	k exp /k ADO	d
	5 -94	dry	4.90 ± 1.70	0.021
	75	11.4	4.55 ± 1.02	0.020
	75	15.5	3.53 ± 0.41	0.015
	5 -96	20.1	2.54 ± 1.57	0.011
	5 -95	21.5	2.69 ± 2.37	0.012
	5 -96	22.6	2.05 ± 0.40	0.008
	75	23.3	2.97 ± 0.54	0.013
	75	30.5	2.55 ± 0.22	0.011
	75	36.5	2.34 ± 0.25	0.010
	75	62.8	1.79 ± 0.20	0.008

a Units: [CH 3 I] saturator : ppbV, k exp : 10 -11 cm 3 s -1 , b Experimental uncertainty in %RH values is ±0.5%RH c Uncertainty is ±2σ, precision only

  3 O + and CH 3 I are summarized in Table 1. The experimental values range from 1.79 × 10 -11 cm 3 s -1 to 4.90 × 10 -11 cm 3 s -1 as a function of relative humidity. On the other hand k ADO = 2.25 × 10 -9 cm 3 s -1 . The reaction efficiency k exp /k ADO is calculated to range from 0.008 (62%RH) to 0.021 in dry nitrogen flow. In this work, k exp /k ADO is equal to [CH 3 I] saturator /[CH 3 I] PTR-MS

  do not list their electric field strength (E/N) values nor which collision rate constant they calculate nor the polarizability nor the dipole moment values, it is difficult to compare the two studies. It is known that the kinetics of the ion-molecule reactions

	are controlled by drift-tube temperature and pressure, drift tube E/N, reactant molecule dipole moment
	and polarizability.

Acknowledgement

This research was supported by the Mitigation of Releases to the Environment (MiRE) project launched by the Institut de Radioprotection et de Sûreté Nucléaire (IRSN) and funded by the the French National Research Agency (ANR) under the convention number 11-RSNR-0013. We gratefully acknowledge this support.