Vincent Pouthier

Quantum decoherence in finite size exciton-phonon systems

Keywords:

Based on the operatorial formulation of the perturbation theory, the properties of a confined exciton coupled with phonons in thermal equilibrium is revisited. Within this method, the dynamics is governed by an effective Hamiltonian which accounts for exciton-phonon entanglement. The exciton is dressed by a virtual phonon cloud whereas the phonons are clothed by virtual excitonic transitions. Special attention is thus paid for describing the time evolution of the excitonic coherences at finite temperature. As in an infinite lattice, temperature-enhanced quantum decoherence takes place. However, it is shown that the confinement softens the decoherence. The coherences are very sensitive to the excitonic states so that the closer to the band center the state is located, the slower the coherence decays. In particular, for odd lattice sizes, the coherence between the vacuum state and the one-exciton state exactly located at the band center survives over an extremely long time scale. A superimposition involving the vacuum and this specific one-exciton state behaves as an ideal qubit insensitive to its environment.

I. INTRODUCTION

Molecular lattices show regularly distributed atomic subunits along which the energy of a specific electronic transition, or a high frequency vibrational mode, delocalizes due to dipole-dipole interaction. This gives rise to a narrow-band exciton [START_REF] May | Charge and Energy Transfer Dynamics in Molecular Systems[END_REF][START_REF] Agranovich | Excitations in Organic Solids[END_REF][START_REF] Bittner | Quantum Dynamics: Applications in Biological and Materials Systems[END_REF] whose quantum states are superimpositions of local states and which is able to move coherently along the lattice. However, the exciton interacts with the phonons of the host medium [START_REF] Fröhlich | [END_REF][5][6] . They induce stochastic modulations of the local state energies and they tend to break the excitonic coherence.

The exciton-phonon system is a prototype of an open system (exciton) coupled with a thermal bath (phonons) whose characterization is essential in understanding many phenomena: exciton dynamics in photosynthetic antenna and polymers [7][8][9][10][11] , vibron propagation in α-helices [START_REF] Scott | Nonlinear Science: Emergence and Dynamics of Coherent Structures[END_REF][START_REF] Edler | [END_REF][14][15][16][17][18][19] and adsorbed nanostructures [20][21][22][23][24][25] , vibrational qubits in quantum channels 26,27 . The exciton properties are thus encoded in the reduced density matrix (RDM) whose behavior is governed by a generalized master equation (GME) [START_REF] Schlosshauer | Decoherence and the Quantum-to-Transition[END_REF][START_REF] Breuer | The theory of open quantum systems[END_REF] . Among the different strategies elaborated to derive a GME, the time-convolutionless approach (TCL) plays a central role [START_REF] Breuer | The theory of open quantum systems[END_REF][START_REF] Shibata | [END_REF][31][32][33][34] . It provides a time local GME in which the phonon influence is encoded in a time-dependent relaxation operator. It is particularly suitable for describing non-markovian processes and for deriving approximate GME. The Born approximation is often invoked resulting in a second order expansion (TCL 2 ) with respect to the exciton-phonon coupling.

In a recent series of papers [35][36][37][38][39] , special attention has been paid for studying a narrow-band exciton coupled with acoustic phonons. Assuming that the exciton propagates slower than the phonons, the nonadiabatic weak coupling limit has been considered (see for instance Refs. [40][41][42] and the references inside). Therefore, the TCL 2 method has been used for describing the evolution of the excitonic coherences. These coherences measuring the ability of the exciton to develop superimpositions involving the vacuum and one-exciton states, they are required to characterize various processes (optical response [START_REF] Mukamel | Principles of Nonlinear Optical Spectroscopy[END_REF] , quantum state transfer [START_REF] Bose | [END_REF] ...). Our studies have revealed that the behavior of the coherences strongly depends on the lattice size.

In an infinite lattice 35,36 , the phonons behave as a reservoir and the Born approximation is legitimate. In the nonadiabatic limit, the Markov limit is reached and the relaxation operator becomes time independent. Dephasing limited-coherent dynamics takes place so that the coherences irreversibly decrease with time. They localize and do not propagate significantly along the lattice 36 . By contrast, in a finite size lattice, a strong non-markovian regime takes place due to the discrete nature of the phonon energy spectrum 37,38 . The confinement favors quantum recurrences that provide to the relaxation operator an almost periodic nature which mainly results from the coupling with the lowest frequency phonon mode (LFPM). The GME reduces to a linear system of differential equations with almost periodic coefficients. Therefore, parametric resonances between specific excitonic frequencies and LFPM give rise to an exponential growth of the RDM indicating that the TCL 2 method breaks down 38 .

To overcome this difficulty, finite size effects have been revisited within the standard perturbation theory (PT) 39 . To proceed, the system "exciton+LFPM" has been considered. Due to its simplicity, it was solved exactly so that the PT performance has been checked. It has been shown that PT is a powerful tool for describing the spectral properties of the system. Furthermore, contrary to TCL 2 , PT is particularly suitable for characterizing the coherence dynamics. Quite naturally, the present paper is thus devoted to the generalization of PT to include the influence of all the phonon modes. However, we shall consider the operatorial formulation of PT that in-volves a perturbative unitary transformation [START_REF] Wagner | Unitary Transformations in Solid State Physics[END_REF] . Within this method, the exciton-phonon system is governed by an effective Hamiltonian that accounts for excitonphonon entanglement. It is no longer necessary to derive the perturbed eigenstates to evaluate the coherences, the required information being encoded in the transformation.

The paper is organized as follows. In Sec. II, the exciton-phonon Hamiltonian is described. Then, PT is applied to build the transformation and to derive the effective Hamiltonian. An approximate expression for the coherences is finally established. In Sec. III, the coherence dynamics is studied numerically and the results are discussed in Sec. IV.

II. THEORETICAL BACKGROUND

A. Model Hamiltonian

In a confined lattice with N sites x = 1, ..., N , oneexciton states are the first excited states of N coupled two-level systems [35][36][37][38][39] . They correspond to N superimpositions of incident and reflected plane waves with quantized wave vectors K k = kπ/L, with k = 1, .., N and L = N + 1. The kth eigenstate is defined as

|k = N x=1 2 L sin (K k x) |x , ( 1
)
where |x is the first excited state of the xth two-level system. The discrete eigenenergies ω k = ω 0 + 2Φ cos(K k ) belong to a band centered on the Bohr frequency ω 0 of each two-level system. Its bandwidth 4Φ involves the exciton hopping constant Φ between neighboring sites ( = 1 will be used throughout this paper). The exciton Hamiltonian is thus defined as :

H A = N k=1 ω k |k k|.
Note that the vacuum state |⊘ describing all the twolevel systems in their ground state with zero energy, the dimension of the exciton Hilbert space E A is equal to N + 1. The phonons are the elementary excitations associated to the external motions of the lattice sites that behave as point masses M connected via force constants W . They refer to N normal modes with wave vectors Q p = pπ/L and frequencies Ω p = Ω c sin(Q p /2), with p = 1, .., N and Ω c = 4W/M . In the phonon Hilbert space E B , the dynamics is governed by the Hamiltonian

H B = N p=1 Ω p a †
p a p , a † p and a p being standard phonon operators. Finally, the exciton-phonon coupling V accounts for the modulation of each two-level system Bohr frequency induced by the lattice vibrations as (for more details see Ref. 38 )

V = N p=1 M p (a † p + a p ), (2)
where the operator M p acts in E A , only. It is defined as

M p = N k=1 N k ′ =1 η p S pkk ′ |k k ′ |. (3)
where

η p = [(E B Ω p /L)(1 -(Ω p /Ω c ) 2 )] 1/2
measures the coupling strength with the pth phonon mode, E B being the small polaron binding energy. In Eq.( 3), S pkk ′ is written as

S pkk ′ = δ p,k-k ′ + δ p,k ′ -k -δ p,k+k ′ -δ p,2L-k-k ′ . ( 4
)
The finite size exciton-phonon system is governed by the Hamiltonian H = H 0 + V where H 0 = H A + H B is the unperturbed Hamiltonian. Since H conserves the exciton number, the Hilbert space E = E A ⊗ E B is partitioned into independent subspaces as E = E 0 ⊕ E 1 . In the zero-exciton subspace E 0 , V = 0 so that the unperturbed states are eigenstates of H. They correspond to tensor products involving the vacuum |⊘ and the phonon number states |{n p } = |n 1 , ..., n N . They describe n p free phonons with energy n p Ω p , p = 1, ..., N . In the oneexciton subspace E 1 , the unperturbed states |k ⊗ |{n p } refer to free phonons accompanied by an exciton in state |k . Since V turns on in E 1 , they are no longer eigenstates of H. The exact eigenstates are entangled excitonphonon states that result from scattering processes which mix both exciton and phonon degrees of freedom. Indeed, V yields exciton scattering from |k with energy ω k , to |k ′ with energy ω k ′ , via the exchange of a phonon p with energy Ω p . The allowed transitions are specified by the rules S pkk ′ = 0 that generalize the concept of momentum conservation in a finite size lattice.

The key point it that these allowed transitions do not conserve the energy within the nonadiabatic limit, as in infinite lattices [START_REF] Silbey | [END_REF][47][48] . Provided that 4Φ < Ω c , there is no resonance between coupled unperturbed states since ω k -ω k ′ = ±Ω p . Consequently, within the weak coupling limit (E B ≪ Φ), second order PT can be applied to treat the influence of the coupling V 39 .

B. Perturbation theory

The operatorial formulation of PT [START_REF] Wagner | Unitary Transformations in Solid State Physics[END_REF] is detailed in Appendix A. It is based on the introduction of a unitary transformation U that diagonalizes the Hamiltonian Ĥ = U HU † in the unperturbed basis. The key point it that U is expanded as a Taylor series with respect to V so that the diagonalization is obtained at a given order. After simple algebraic manipulations, the transformed Hamiltonian up to second order is written as

Ĥ = ĤA + N k=1 Ĥ(k) B ⊗ |k k| (in E 1 ), where ĤA = N k=1 (ω k + δω k )|k k| Ĥ(k) B = N p=1 (Ω p + δΩ pk )a † p a p . (5)
The contribution ĤA is the effective exciton Hamiltonian. It reveals that in state |k , the energy of an exciton ωk = ω k + δω k is renormalized due to its coupling with the phonons. The correction δω k is defined as (Eq.(A8))

δω k = N p=1 N k ′ =1 k|M p |k ′ 2 ω k -ω k ′ -Ω p . (6)
It results from the spontaneous emission of a phonon p during which the exciton realizes a transition from |k to |k ′ . However, in the nonadiabatic limit, the energy is not conserved during the transition so that the phonon emission is not a real process. The exciton is only able to exchange a virtual phonon which is first emitted and then immediately reabsorbed. In accordance with the small polaron concept, δω k is the manifestation of the so-called dressing effect. The exciton does no longer propagate freely but it is dressed by a virtual phonon cloud. Similarly, when the exciton occupies a state |k , the phonon dynamics is governed by an effective Hamiltonian Ĥ(k) B (Eq.( 5)). Each phonon of the pth mode experiences an energy shift δΩ pk expressed as (Eq.(A8))

δΩ pk = N k ′ =1 2 k|M p |k ′ 2 (ω k -ω k ′ ) (ω k -ω k ′ ) 2 -Ω 2 p . (7)
The correction δΩ pk results from two mechanisms. First, the phonon p can be absorbed giving rise to excitonic transitions from |k to |k ′ . Such a process does not conserve the energy within the nonadiabatic limit so that the phonon is immediately re-emitted. Second, the phonon p can induce the stimulated emission of a second phonon during which the exciton realizes transitions. But, as previously, the emitted phonon is immediately reabsorbed. Both mechanisms are virtual processes indicating that the phonons do no longer evolve freely but are dressed by virtual excitonic transitions. The operator Ĥ is the effective exciton-phonon Hamiltonian. Being diagonal in the unperturbed basis, its eigenvalues define the system eigenenergies up to second order in V as ǫ k,{np} = ωk + N p=1 n p (Ω p + δΩ pk ). The corresponding eigenstates are defined as |Ψ k,{np} = U † |k ⊗ |{n p } . Consequently, U provides a new point of view in which Ĥ does no longer describe independent excitations but refers to entangled exciton-phonon states. In this new point of view, a state |k defines an exciton dressed by a virtual phonon cloud whereas the number state |{n p } describes phonons clothed by virtual excitonic transitions.

As shown in the next section, the new point of view is particularly suitable for deriving an approximate expression for the excitonic coherences. However, it is expected to give good results provided that the coupling between unperturbed states is smaller than the corresponding Bohr frequencies. By studying the different coupling paths, it turns out that the LFPM is responsible for the largest perturbation. Consequently, as shown previously 39 , PT can be applied at temperature T provided that (k B is the Boltzmann constant)

4E B k B T π 2 Ω 2 c L < x, (8)
where x ≈ 0.05 specifies the desired accuracy for the approximate eigenenergies. For a given temperature and a fixed coupling strength, Eq.( 8) shows that a critical length L * discriminates between two regimes. For L < L * , PT correctly describes the exciton-phonon dynamics whereas it certainly breaks down for L > L * .

C. Excitonic coherences

Without any perturbation, the lattice is assumed to be in thermal equilibrium at temperature T . We thus consider situations for which ω 0 ≫ k B T so that the exciton cannot be thermally excited. By contrast, the phonons form a thermal bath whose properties are described by the density matrix

ρ B = exp(-βH B )/Z B , Z B being the phonon partition function (β = 1/k B T ).
To study the exciton dynamics, the lattice is brought in a configuration out of equilibrium in which the exciton is prepared in a state |ψ A = |⊘ [35][36][37][38][39] . This step is supposed to be rather fast when compared with the typical time evolution of the phonons. As a result, the initial density matrix is defined as

ρ(0) = |ψ A ψ A | ⊗ ρ B .
Of course, |ψ A depends on the properties under study. To analyze energy transport, one can choose |ψ A = |x 0 indicating that an excitonic population is created on the x 0 th site [START_REF] Silbey | [END_REF][47][48][49][50] . In the present approach, we consider that |ψ A is a superimposition involving the vacuum |⊘ and the one-exciton eigenstate |k

1 as |ψ A = c 0 |⊘ + c 1 |k 1 , with |c 0 | 2 + |c 1 | 2 = 1.
Note that such a superimposition may result from an optical excitation of the lattice.

In that context, the exciton properties are encoded in the RDM σ(t) = T r B [ρ(t)], where T r B is a partial trace over the phonon degrees of freedom. Because H conserves the exciton number, σ(t) exhibits independent blocks. Matrix elements σ xx (t) and σ xx ′ (t) define the one-exciton block that is required to study the transport properties [48][49][50] . Here, we restrict our attention to the block containing the coherences written as

σ k2⊘ (t) = G k2k1 (t)σ k1⊘ (0) (σ k1⊘ (0) = c * 0 c 1 ) with G k2k1 (t) = k 2 |T r B ρ B e iHB t e -iHt |k 1 . (9)
The coherence σ k2⊘ (t) measures the ability of the exciton to develop a superimposition involving |⊘ and |k 2 at time t, given that a superimposition involving |⊘ and |k 1 was initially created. Consequently, G k2k1 (t) generalizes the concept of transition amplitude and it defines the effective exciton propagator. It yields the probability amplitude to observe the exciton in |k 2 at time t given that it was in |k 1 at t = 0. More precisely, the full system being prepared in the factorized state |k 1 ⊗ |{n p } , G k2k1 (t) measures the probability amplitude to observe the system in a factorized state exp(-i p n p Ω p t)|k 2 ⊗|{n p } . It thus describes an excitonic transition during which the phonons evolve freely, an average over the initial phonon state being performed.

The evaluation of Eq.( 9) within PT is detailed in Appendix B. To proceed, we first introduce U and diagonalize H. Second, we use the fact that Ĥ is the sum of independent contributions, each contribution describing the exciton-phonon system when the exciton occupies a specific state |k . Then, we define the following Heisenberg representation

O k (t) = e i Ĥ(k) B t Oe -i Ĥ(k) B t
. Finally, we introduce the effective density matrix

ρ (k) B (t) = 1 Z (k) B (t) N p=1 exp -(βΩ p + iδΩ pk t)a † p a p , (10)
where

Z (k) B (t) = N p=1 1 1 -exp [-(βΩ p + iδΩ pk t)] . (11)
Strictly speaking, ρ

B (t) is not a density matrix since it provides complex values for the population of the phonon number states. However, it reduces to the equilibrium density matrix at t = 0, i.e. ρ (k)

B (t = 0) ≡ ρ B ∀k. More- over, ρ (k)
B (t) being isomorphic to ρ B , the trace over the phonon degrees of freedom will provide averages equivalent to thermal averages but with the correspondence βΩ p → βΩ p + iδΩ pk t.

In that context, after simple algebraic manipulations, the effective propagator is rewritten as (see Appendix B)

G k2k1 (t) = N k=1 Z (k) B (t) Z B exp[-i(ω k + δω k )t] × (12)
T r B ρ

(k) B (t) k 2 |U † k (t)|k k|U k (0)|k 1 .
Expanding U in a Taylor series with respect to V , one finally obtains the second order expression of the effective exciton propagator displayed in Appendix C (Eq.(C1)).

III. NUMERICAL RESULTS

In this section, the previous formalism is used for describing excitonic coherences in a finite size lattice. In spite of its general nature, it will be applied to amide-I exciton in α-helices, a system extensively studied in our previous works [35][36][37][38][39] . In α-helices, peptide units linked by H bonds are regularly distributed. Each unit contains an amide-I mode (high frequency C=O vibration) that gives rise to vibrational excitons coupled with the phonons of the H bond network. This system is a prototype of a finite size lattice since helices containing from 3 to 15 residues are the most abundant in nature 51 . Of course, the present model is too simple to accurately describe vibrational energy flow in real helices whose dynamics exhibits a tremendous complexity due to the large number of degrees of freedom. Nevertheless, it involves ingredients that play a key role in interpreting specific experiments such as pump-probe spectroscopy in α-helices [START_REF] Edler | [END_REF] and Electron Capture Dissociation in finite size polypeptides 18 . Unfortunately, this model is unable to describe energy relaxation because it conserves the number of amide-I exciton. In proteins, the amide-I lifetime, typically of about 1 ps, results from intramolecular energy redistribution due to the anharmonic coupling between each amide-I mode and a set of intramolecular normal modes whose displacements are strongly localized on the C=O groups [52][53][54] . Consequently, we do not claim that the model is relevant to explain in details the vibrational dynamics in a protein. Nevertheless, its interest lies in the fact that it provides a simple approach to promote the idea that PT is more efficient than TCL 2 method which has been applied recently to a similar model 38 .

Typical values for the parameters are used: ω 0 = 1660 cm -1 , W = 15 Nm -1 , M = 1.8 × 10 -25 kg, Ω c = 96.86 cm -1 and Φ = 7.8 cm -1 . The adiabaticity B = 2Φ/Ω c is equal to 0.16 so that the nonadiabatic limit is reached. The coupling strength will be discussed in terms of the χ parameter usually introduced in the Davydov model (E B = χ 2 /W ). It will vary around χ = 8 pN, a value close to ab initio estimate [START_REF] Ostergard | Davydov Soliton Revisited[END_REF] . With these parameters, the critical length L * ≈ 20 ensures the PT validity over the temperature range T ∈ [0, 300] K (see Eq.( 8)).

As shown in Fig. 1a, the phonons induce a red-shift of each excitonic eigenenergy that scales as δω k ≈ -E B . Nevertheless, δω k depends on both k and N . A small dispersion arises and δω k increases with k (Φ > 0). For N = 10, δω 1 = -0.99E B whereas δω 10 = -0.71E B . Consequently, δω k defines a band centered on a mean value δ ω and whose width is about ∆ω = δω N -δω 1 . It turns out that δω k rapidly decreases with L and it finally converges. Such a behavior is encoded in the size dependence of the band center that scales as δ ω ≈ -1.056E B (1 -1.96/L). Nevertheless, since all the corrections behave similarly, the bandwidth is less sensitive to the lattice size. It scales as ∆ω ≈ (0.28 -2.48/L 2.86 )E B and it is about 0.28E B provided that L > 7.

As displayed in Figs. 1b and1c, the energy of each phonon mode is either red-shifted or blue-shifted due to the dressing by virtual excitonic transitions. In a general way, an exciton whose energy is above ω 0 yields a red-shift of the phonon energy whereas an exciton whose energy is below ω 0 favors a blue-shift. Nevertheless, opposite situations may appear, especially when the exciton occupies states close to the band center. For N = 8, the exciton in state k = 4 produces a blue-shift of the frequency of the phonon mode p = 5 whereas the exciton in state k = 5 yields a red-shift. In addition, specific phonon modes remain insensitive to the exciton that occupies a particular state. For N = 8, when the exciton is either in the state k = 3 or k = 6, the phonon mode p = 6 is not perturbed. As shown in Fig. 1c, a remarkable effect arises for odd N values. In that case, phonons dressed by an exciton whose energy is exactly located at the band center are not perturbed. For k = L/2, one obtains δΩ pk = 0 ∀ p. Note that the intensity of δΩ pk decreases with the lattice size. Phonon modes with p ≈ L/2 exhibit the largest energy shift whereas phonon modes with p ≈ N seem to be less affected.

Let us now study the time evolution of the excitonic coherences. To proceed, we shall characterize the diagonal elements of the effective propagator, only. We have verified that such a restriction is valid because the propagator is diagonally dominant in the nonadiabatic weak coupling limit. Therefore, to simplify the notation, G kk (t) will be called the coherence of the state k. For an even lattice size (N = 8), the time evolution of |G kk (t)| is illustrated in Fig. 2a for T = 100 K. Over a short time scale (t < 100 ps), |G kk (t)| shows high frequency small amplitude fluctuations just below its initial value equal to unity (inset in Fig. 2a). Then, it follows a smooth decaying function that supports a high frequency small amplitude modulation indicating that quantum decoherence takes place. Such a behavior is strongly k dependent. Indeed, the coherences evolve in pairs since

|G kk (t)| ≈ |G L-kL-k (t)|.
Then, the coherence of states close to the band edges (ω k ≈ ω 0 ± 2Φ) decays faster than the coherence of states located near the band center (ω k ≈ ω 0 ). For instance, at t = 100 ps, |G 11 (t)| = 0. behave as in Fig. 2. They still evolve in pairs and they first exhibit fast oscillations just below their initial value. As time increases, they follow smooth decaying functions that show high frequency small amplitude modulations. Quantum decoherence occurs and, as previously, it is enhanced by the temperature. Nevertheless, few differences take place. First, the modulation amplitude appears larger, especially for states located close to the band center. Then, the coherences that decay do no longer vanish in the long time limit. This is particularly pronounced at high temperature (Fig. 3b) for which the coherences converge to a finite value by exhibiting high frequency oscillations. The closer to the band center the state is located, the larger the asymptotic coherence value and the oscillation amplitude are. However, a surprising effect takes place that is absent for even N values. Indeed, the coherence of the state exactly located at the band center (k = L/2) behaves differently. In the short time limit, it slightly decreases from unity by exhibiting high frequency damped oscillations. As time increases, the oscillations disappear and the coherence converges to a constant value C L/2 rather close to unity. The influence of the temperature is twofold. First, as T increases, C L/2 decreases. Then, the amplitude of the damped oscillations increases with T . Nevertheless, the larger the temperature is, the faster the modulation disappears. As illustrated in Fig. 4 for N = 8 and T = 100 K, quantum decoherence is not irreversible and specific features arise in the long time limit. Indeed, each coherence decreases until it recurs at specific revival times. For it increases again over a time scale which extends from 2900 ps to 4400 ps. Then additional recurrences occur around 5800 ps, 7300 ps and 8700 ps, the amplitude of which being smaller than 0.08. However, we have verified the recurrences decrease as T increases so that they almost disappear at room temperature. For instance, for N = 8, T = 300 K and k = 1, recurrences still arise but their amplitudes remain smaller than 0.001. Similarly, the larger N is, the smaller the recurrences are.

By contrast, even over very long time scales, the coherence of the state k = L/2 survives and it rapidly converges to C L/2 . As shown in Fig. 5a, C L/2 decreases almost linearly as T increases. A nonlinear dependence arises at very low temperature, only. For N = 9 and χ = 8 pN, C L/2 = 0.95 for T = 100 K whereas C L/2 = 0.85 for T = 300 K. Moreover, the coupling enhances the temperature effect. Similarly, C L/2 is a linear function of the lattice size (Fig. 5b). For T = 300 K and χ = 8 pN, it decreases from 0.92 to 0.71 when N varies between 5 and 17. Finally, as shown in Fig. 5c, C L/2 also decreases linearly with E B and it decays from 0.85 to 0.67 when E B varies between 0.21 cm -1 (χ = 8 pN) and 0.47 cm -1 (χ = 12 pN). These results reveal that the asymptotic value of the coherence of the state k = L/2 scales as depending on the lattice size parity. For instance, for N = 8, the decoherence rate extends from Γ 4 = 0.028 cm -1 to Γ 1 = 0.099 cm -1 . By contrast, for N = 9, it ranges between Γ 5 = 0 and Γ 1 = 0.096 cm -1 . Note that we have verified that Γ 1 decreases with the lattice size and typically scales 1/ √ L. As illustrated in Fig. 6b for N = 9 and χ = 8 pN, Γ k increases linearly with the temperature. Note that a nonlinear dependence arises at very low temperature, only. However, such a behavior strongly depends on the excitonic state. First of all, Γ L/2 = 0 ∀T . Then, the closer to the band edge the state is located, the faster Γ k increases with the temperature. For instance, Γ 1 varies between 0.031 cm

C L/2 = 1 -α(N -1), with α ≈ 3.82E B k B T /Ω 2 c . Let us finally characterize the decoherence rate Γ k . It is defined as Γ k = 1/T k ,

IV. DISCUSSION

In a confined environment, the coherence dynamics is controlled by different time scales. Over a short time scale, the coherences fluctuate just below their initial value equal to unity. Then, quantum decoherence takes place and each coherence follows a smooth decaying function that supports a small amplitude modulation. After a time scale specified by the decoherence time, the coherence either vanishes or exhibits small amplitude oscillations close to zero, depending on the parity of the lattice size. Finally, coherence revivals occur in the long time limit defining small recurrences whose amplitude decreases with both T and N . The main observation is that quantum decoherence strongly depends on the exciton eigenstate. The closer to the band center the state is located, the longer the decoherence time is. In particular, for odd lattice sizes, the coherence of the state exactly located at the band center survives over an infinite time scale, even at room temperature.

To interpret these features, let us first mention that within PT, the influence of the exciton-phonon interaction is encoded into two main parameters. The first parameter, δω k , describes the energy shift experienced by the exciton when it is dressed by a virtual phonon cloud. According to the small polaron concept 41,42 , this dressing yields a red-shift of the Bohr frequency of each two-level system. This shift, equal to E B in an infinite lattice, reduced to E B (1 -2/L) in a confined environment 42 . Moreover, the dressed exciton delocalizes according to a reduced hopping constant Φ ≈ Φ exp(-S), where the band-narrowing factor at zero temperature is S = 8E B /(3πΩ c ) 41 . Note that in a finite size lattice Φ is both inhomogeneous and size dependent [START_REF] Falvo | [END_REF] . But these effects remain weak in a lattice with fixed boundary conditions. As a result, the energy corrections behave as

δω k ≈ -E B (1 -2/L) -2ΦS cos(kπ/L). ( 13
)
As observed in Fig. 1, δω k forms a band centered on δ ω = -E B (1-2/L) and whose width ∆ω = 4ΦS reduces to 0.273E B with the parameters used in the figure.

The second parameter accounts for the fact that, dressed by the virtual transitions of an exciton in state k, the energy of a phonon p becomes Ωp = Ω p + δΩ pk . To evaluate energy correction δΩ pk displayed in Figs. 1b and1c, let us approximate Eq.( 7) by considering normal scattering processes k ′ = k -p and k ′ = k + p, only. Up to first order in the adiabaticity, one obtains

δΩ pk ≈ - 8BE B L Ω p Ω c 1 - Ω p Ω c 2 cos kπ L . (14)
Eq.( 14) shows that δΩ pk < 0 if k < L/2 whereas δΩ pk > 0 if k > L/2. It reveals that δΩ pk = 0 ∀ p for k = L/2. The most perturbed phonon mode satisfying Ω p = Ω c / √ 3, the maximum energy shift is 3.08BE B /L, i.e. about 5E B /100 for L = 9 -10, as observed in Fig. 1.

In that context, assuming that the system eigenstates reduce to the unperturbed states, the effective exciton propagator is rewritten as (see Eq.(C2) with Λ

(2) k = 0) G kk (t) ≈ exp(-iω k t)Z (k) B (t)/Z B .
The corresponding modulus is thus defined as

|G kk (t)| ≈ N p=1 1 1 + 4∆n 2 p sin 2 (δΩ pk t/2) , (15)
where ∆n 2 p = np (n p + 1) measures the thermal fluctuations of the pth phonon number around its average value np = [exp(βΩ p ) -1] -1 . Although it is quite simple, Eq.( 15) provides a clear understanding of the different features observed in the previous section. First of all, since δΩ pL-k = -δΩ pk (see Eq.( 7)), it turns out that |G kk (t)| = |G L-kL-k (t)|, as observed in Figs. 2, 3 and4.

Then, Eq.( 15) describes a smooth decaying function that tends to zero over intermediate time scale, revealing that quantum decoherence occurs. The physics of the decoherence can be understood as follows. At t = 0, the system is prepared in a factorized state |Ψ(0) = [c 0 |⊘ + c 1 |k ] ⊗ |φ . This state describes an exciton in a superimposed state accompanied by phonons in a well defined number state |φ ≡ |{n p } , phonons and exciton being independent. As time increases, this state evolves in an entangled exciton-phonon state as

|Ψ(t) ≈ c 0 |⊘ ⊗ |φ ⊘ (t) + c 1 e -iω k t |k ⊗ |φ k (t) , (16)
where |φ ⊘ (t) is the state of free phonons whereas |φ k (t) describes the state reached by dressed phonons. Building the system density matrix and performing a trace over the phonon degrees of freedom, one obtains G kk (t) ≈ exp(-iω k t) φ ⊘ (t)|φ k (t) . The phase factor, that does not affect the coherence dynamics, accounts for the evolution of the dressed exciton. By contrast, the inner product φ ⊘ (t)|φ k (t) defines the so-called decoherence function that measures the ability of the phonons to evolve freely in spite of their coupling with the exciton. This object is rather general in the theory of open quantum systems [START_REF] Schlosshauer | Decoherence and the Quantum-to-Transition[END_REF][START_REF] Breuer | The theory of open quantum systems[END_REF] . It suggests that quantum decoherence results from the fact that the bath evolves differently as the open system occupies different quantum states (see for instance Ref. 57 ). In the present situation, a slightly different interpretation emerges. Indeed, when the phonons are initially in a pure state, the inner product reduces to a phase factor p exp(in p (Ω p -Ωpk )t) that involves the energy difference between free and dressed phonons. Although this phase factor accounts for the fact that dressed phonons evolve differently when compared with bare phonons, it does not affect the excitonic coherence. However, at finite temperature, the phonons are described by a statistical mixture of number states so that an average procedure is required. Therefore, performing the average over the initial phonon state yields a sum over phase factors which interfere some with the others, resulting in the decay of the excitonic coherence.

As shown in Eq.( 15), describing the coherence decay remains a hard task since |G kk (t)| involves the product of periodic functions. To overcome this difficulty, we first express the coherence as |G kk (t)| = exp[-F k (t)], F k (t) being the decoherence factor. Expanding F k (t) in the short time limit, one finally obtains

|G kk (t)| ≈ exp - 1 2 N p=1 ∆n 2 p δΩ 2 pk t 2 . (17)
Eq. (17) indicates that the coherence behaves as a Gaussian function whose half width at half maximum yields an approximate expression for the decoherence rate Γ g k . Note that a similar time evolution has been proposed to characterize quantum decoherence in a solute/bath system 57 . However, we have verified that |G kk (t)| scales as a Gaussian function provided that t < 1/Γ g . Over longer time scale, a slower decay takes place. Consequently, Γ g k always over estimates the exact decoherence rate displayed in Fig. 6. However, after a detailed analysis of our numerical data, it turns out that the ratio Γ k /Γ g k only depends on the lattice size. Therefore, a quite good estimate of the decoherence rate that clearly reproduces the data displayed in Fig. 6 is expressed as

Γ k ≈ 1 -1/L √ 2 ln 2 N p=1 ∆n 2 p δΩ 2 pk . (18)
As shown in Eq.( 18), the temperature dependence of Γ k is encoded in the fluctuations of the phonon numbers. Since ∆n p typically scales as k B T /Ω p , Γ k increases linearly with the temperature. Note that this dependence differs from the standard expression of the dephasing rate that characterizes the decoherence of an open system coupled with a reservoir of harmonic oscillators 1 . In that case, GME methods show that the temperature dependence of the rate originates in its dependence with respect to the average phonon number np . At high temperature, both approaches yield a similar temperature dependence since ∆n p ≈ np ≈ k B T /Ω. This is no longer the case at low temperature since ∆n p /n p = exp(βΩ p /2). Moreover, Γ k is proportional to E B through its dependence with respect to the phonon energy corrections (see Eq.( 7)).

From this dependence, we recover that the closer to the band edges the excitonic state is located, the larger Γ k is.

As observed in Fig. 3, Eq.( 18) also reveals that the coherence of the state located at the band center (k = L/2) survives over an infinite time scale. This feature occurs for an odd lattice size only because δΩ pL/2 = 0 ∀p. Indeed, during both phonon absorption and phonon stimulated emission, the virtual excitonic transition form |k to |k ′ is counterbalanced by the transition form |k to |L -k ′ . Due to the symmetry of the excitonic energy spectrum ((ω ′ k -ω 0 ) = -(ω L-k ′ -ω 0 )) and because M pkk ′ = M pkL-k ′ , both contributions exactly compensate when k = L/2. As a result, a zero phonon energy shift occurs so that Γ L/2 → 0. Note that the asymptotic behavior of the coherence C L/2 is not described by Eq. (15). As shown in Fig. 5, C L/2 decreases linearly with both T , E B and N . Such a behavior originates in the second order correction of the eigenstates that ensure their normalization to unity. As shown in Eq.(C2), this correction yields the contribution Λ

(2) k (t) which provides to C L/2 its dependence with respect to the model parameters.

Finally, Eq.( 15) gives formation about the long time behavior of the coherences. Indeed, when restricting the influence of the phonon bath to the pth mode, only, |G kk (t)| reduces to a periodic function of time with period τ pk = 2π/|δΩ pk |. As a result, although the coherence decays due to quantum decoherence, it finally recurs periodically at specific revivals time τ pk , 2τ pk , 3τ pk ... and so on. These revivals time define exact recurrences for which the coherence reaches unity. However, this ideal scenario breaks down when all the phonon modes are considered. The main reason it that it is unlikely that the contribution of each phonon mode will recur simultaneously to provide a strong recurrence. In other words, it is as if the coherence revivals due to the coupling with a particular phonon mode were screened by the remaining modes. Nevertheless, each phonon mode yields specific features that can be distinguished in the long time behavior of the coherence at low temperature and in short lattices. For instance, with the parameters used in Fig. 4, the coherence of the state k = 1 slightly recurs over a time scale which extends from 2900 ps to 4400 ps. This feature is the signature of coherence revivals that take place at the revival times τ 11 = 4357 ps, τ 21 = 3655 ps, τ 31 = 2919 ps, τ 41 = 2873 ps and τ 51 = 3398 ps. Moreover, the observed recurrences around 5800 ps, 7300 ps and 8700 ps are located at the revival times 2τ 31 , 2τ 21 and 2τ 11 , respectively.

To conclude, let us discuss the implications of the present work for quantum information processing in condensed matter. Indeed, quantum state transfer (QST) from one region to another is a fundamental task in quantum computing 58 . Over short length scale, to ensure communication inside a computer or between adjacent computers, solid-state based system is the ideal candidate for scalable quantum computing. However, the quantum channel strongly depends on the way the information is encoded. Since it has been suggested that qubits may be encoded in high frequency vibrational modes [59][60][61] , vibrational exciton-mediated QST is a promising way for quantum information processing 26,27 .

In that context, the confinement of the excitons appears as a key ingredient to fight against quantum decoherence, the public enemy number one in quantum computing. In an infinite lattice, the exciton eigenstates are Bloch states whose coherent nature rapidly disappears due to the coupling with the phonon bath 36 . The coherences decay exponentially according to a state independent decoherence rate of about Γ = 4E B k B T /Ω c . With the parameters used in the present study, the corresponding decoherence time is about 2.86 ps at room temperature. By contrast, the confinement softens the decoher-ence experienced by the exciton, even at high temperature. Although quantum decoherence strongly depends on the excitonic states, the decoherence times typically range between 50 -200 ps at 300 K. Furthermore, it turns out that the excitonic state exactly located at the band center is insensitive to the phonon bath, its decoherence time being infinite. Of course, such a behavior results from a second order PT and high order theory will provide non vanishing Γ L/2 values. Nevertheless, even if quantum decoherence affects this state, one may expect that the decoherence time will be extremely long in the nonadiabatic weak coupling limit. A superimposition involving the vacuum and this particular one-exciton state will be able to keep its coherent nature over an extremely long time scale. At high temperature, such a superimposition can be viewed as an ideal qubit insensitive to quantum decoherence.

V. CONCLUSION

In this paper, the properties of a confined exciton coupled with phonons in thermal equilibrium has been revisited using the operatorial formulation of PT. This method provides a new point of view in which the dynamics is governed by an effective Hamiltonian that does no longer characterize independent excitations but accounts for exciton-phonon entanglement. The exciton is dressed by a virtual phonon cloud whereas the phonons are clothed by virtual excitonic transitions. Within this new point of view, special attention has been paid for describing the excitonic coherences that measure the ability of the exciton to develop superimpositions involving the vacuum and one-exciton states.

In the nonadiabatic weak coupling limit, it has been shown that the coherence behavior results from an average procedure over the phonon degrees of freedom. Indeed, when the phonons occupy a well-defined number state, the decoherence function reduces to a phase factor although dressed phonons behave differently when compared with bare phonons. At finite temperature, the average over the initial phonon state yields a sum over phase factors which interfere some with the others, resulting in the decay of the excitonic coherence. Therefore, although each coherence remains close to unity over a rather short time scale, it finally decreases as time increases. After a time scale specified by the decoherence time, the coherence either vanishes or shows small amplitude oscillations close to zero, depending on the parity of the lattice size. Coherence revivals have been observed in the long time limit, but their amplitude remains extremely small.

The key point is that the confinement softens quantum decoherence. Indeed, it has been shown that the coherences are very sensitive to the excitonic states and to the lattice size. Therefore, the closer to the band center the state is located, the slower the coherence decays. In particular, for odd lattice sizes, the coherence of the state exactly located at the band center survives over an infinite time scale. In that context, it has been pointed out that a superimposition involving the vacuum and this particular one-exciton state behaves as an ideal qubit insensitive to the phonon bath, even at room temperature.

To conclude, let us mention that PT is valid as long as we remain confined in a small region of the parameter space, i.e. in the weak coupling limit and for short lattice sizes. Therefore, a more general framework is required to correctly describe size effects in both intermediate and strong coupling limits. Based on the small polaron concept, these features will be addressed in forthcoming works. In its operatorial formulation [START_REF] Wagner | Unitary Transformations in Solid State Physics[END_REF] , PT is based on the introduction of a unitary transformation U that diagonalizes the transformed Hamiltonian Ĥ = U HU † in the unperturbed basis. It is written as U = exp(S), where S is an anti-hermitian operator assumed to be non-diagonal in the unperturbed basis. It is expanded as a Taylor series as S = S 1 + S 2 + S 3 + ... where S q is the qth order correction in the coupling V . Consequently, Ĥ becomes From Eq.(A1), S is derived order by order to diagonalize Ĥ at the desired order. Up to second order in V , the solution of the problem is given by the following equations

[H 0 , S 1 ] = V [H 0 , S 2 ] = 1 2 [S 1 , V ] nd Ĥ = H 0 + 1 2 [S 1 , V ] d , (A2)
where d and nd define the diagonal part and the nondiagonal part of an operator in the unperturbed basis. Since V is a linear combination of creation and annihilation phonon operators (see Eq.( 2)), S 1 is of the form

S 1 = p Λ p a † p -Λ † p a p . (A3)
The unknown operator Λ p acts in E A , only. No restriction affects this operator since S † 1 = -S 1 . Therefore, inserting Eq.(A3) into Eq.(A2) yields

k|Λ p |k ′ = k|M p |k ′ ω k -ω k ′ + Ω p . ( A4
)
The knowledge of S 1 allows us to compute the commutator [S

(C3)

By contrast, δG kk (t) is the contribution of the propagator that involves the components of the system eigenstates which are directly coupled to the unperturbed states through the interaction V . These components account for a variation of the phonon number equal to unity. The contribution δG kk (t) reduces to the last two terms in Eq.(C1) for k 1 = k 2 .

FIG. 1 :

 1 FIG. 1: (a) Size dependence of the exciton energy correction δω k /EB vs k. (b) Phonon energy correction δΩ pk /EB vs p for N = 8 and for different k values. (c) Phonon energy correction δΩ pk /EB vs p for N = 9 and for different k values.

FIG. 2 :

 2 FIG. 2: Time evolution of the coherence |G kk (t)| for N = 8, χ = 8 pN and (a) T = 100 K and (b) T = 300 K (The inset illustrates the short time behavior for T = 100 K).

FIG. 3 :

 3 FIG. 3: Time evolution of the coherence |G kk (t)| for N = 9, χ = 8 pN and (a) T = 100 K and (b) T = 300 K.

2 FIG. 5 :

 25 FIG. 5: Behavior of the asymptotic value C L/2 of the coherence of the state k = L/2 versus (a) the temperature (N = 9), (b) the lattice size (T = 300 K) and (c) the coupling strength (N = 9).

FIG. 6 :

 6 FIG. 6: Behavior of the decoherence rate Γ k . (a) Γ k vs k for T = 300 K, χ = 8 pN. (b) Γ k vs T for N = 9 and χ = 8 pN. (c) Γ k vs EB for N = 9 and T = 300 K.

Ĥ 2 [S 1 ,

 21 = H 0 + V + [S 1 , H 0 ] + [S 1 , V ] + [S 2 , H 0 ] + 1 [S 1 , H 0 ]] + ... (A1)

  instance, although |G 11 (t)| almost vanishes after 1000 ps,

		0.12		k=1 k=2 k=3 k=4
	kk (t)| |G	0.08		k=5 k=6 k=7 k=8
		0.04
		0.00
		0	2000	4000	6000	8000	10000
				time (ps)
	FIG. 4: Long time evolution of the coherence |G kk (t)| for
	N = 8, T = 100 K and χ = 8 pN.

  1 , V ] that is required to derive both Ĥ and S 2 .|C p |k 1 + k 2 |D pp -D † pp |k 1 n (k2) |C p |k 1 + k 2 |D pp -D † pp |k 1 n (k1) Ωp+δΩ pk )t k 2 |Λ p |k k|Λ † p |k 1 n (k) Ωp+δΩ pk )t k 2 |Λ † p |k k|Λ p |k 1 (n (k) -1 .The diagonal elements of the effective propagator can be partitioned into two contributions as G kk (t) = G 0 kk (t) + δG kk (t). Basically, G kk (t) reduces to G 0 kk (t) when the system eigenstates are assumed to be collinear to the unperturbed states. One thus obtains

			G k2k1 (t) =	Z	(k1) B (t) Z B	e -iω k 1 t δ k2,k1
					-	Z	(k2) B (t) 2Z B	e -iω k 2 t	p	k 2 |Λ p Λ † p |k 1 n (k2) p	(t) + k 2 |Λ † p Λ p |k 1 (n (k2) p	(t) + 1)
					-	Z	(k1) B (t) 2Z B	e -iω k 1 t	p	k 2 |Λ p Λ † p |k 1 n (k1) p	(t) + k 2 |Λ † p Λ p |k 1 (n (k1) p	(t) + 1)
					+	Z	(k2) B (t) Z B	e -iω k 2 t	p	k 2 p	(t)
					-	Z	(k1) B (t) Z B	e -iω k 1 t	p	k 2 p	(t)
					+	k	Z	(k) B (t) Z B	e -iω k t	p	e i(p (t)
					+	k	Z	(k) B (t) Z B	e -iω k t	p	e -i(p (t) + 1),	(C1)
	where n	(k)
			G 0 kk (t) =	Z	(k) B (t) Z B	e -iω k t [1 -Λ	(2) k (t)],	(C2)
	where Λ ensures their normalization up to second order as (2) k (t) arises from the eigenstate correction that
	Λ	(2) k (t) =

p (t) = [exp(βΩ p + iδΩ pk t) -1] p k|Λ p Λ † p |k n (k) p (t) + k|Λ † p Λ p |k (n (k) p (t) + 1).

This commutator is defined as

where

From the diagonal part of Eq.(A5), the transformed Hamiltonian Ĥ reduces to

Within the excitonic eigenbasis, Eq.(A7) yields Eqs.( 5)- (7) with the definitions

From the non diagonal part of Eq.(A5), one seeks S 2 of the form

where the unknown operators E pp ′ , D pp ′ and C p act in E A , only. Note that C † p = -C p to ensure the anti-hermitian nature of S 2 . By inserting Eq.(A9) into Eq.(A2), these operators are finally defined as

where

APPENDIX B: GENERAL EXPRESSION OF THE EXCITONIC COHERENCES

By inserting the unitary transformation U , the coherence Eq.( 9) is rewritten as

Since Ĥ is a sum of independent contribution (see Eq.( 5)), the coherence is expressed as

Because the operator exp(-i Ĥ(k) B t) defines a unitary evolution and since [H B , Ĥ(k) B ] = 0, the partial trace in Eq.( B2) is rewritten as

At this step, let first define U k (t) = e i Ĥ(k) B t U e -i Ĥ(k) B t as the Heisenberg representation of the unitary transformation U with respect to the Hamiltonian Ĥ(k) B . Then, since ρ B = exp(-βH B )/Z B , one obtains

From the expressions of both H B and H

B , Eq.(B3) can be easily evaluated to finally obtain Eqs. (10) and (11). Therefore, inserting Eq.(B3) into the previous expression of the partial trace and combining the results with Eq.(B2) yield the effective propagator Eq. (12).

APPENDIX C: SECOND ORDER EXPRESSION OF THE EXCITONIC COHERENCES

The approximate expression of the effective exciton propagator up to second order in the exciton-phonon coupling V is written as


Antonia C Kastoris

Petros I Rafailidis

Evridiki K Vouloumanou

Ioannis D Gkegkes

Matthew E Falagas
email: m.falagas@aibs.gr

PHARMACODYNAMICS

Keywords: Drug-resistance, Synergistic combinations, Antagonism, MRSA, ESBLs A, C

Synergy of fosfomycin with other antibiotics for

Gram-positive and Gram-negative bacteria

Introduction

The alarmingly increasing drug resistance rates observed in our era limit our armamentarium of antibiotics and necessitate the development and evaluation of alternative ways to cope with this issue. Specifically, the renewal of physicians' interest in older and neglected antibiotic agents [START_REF] Falagas | Potential of old-generation antibiotics to address current need for new antibiotics[END_REF][START_REF] Falagas | Colistin: the revival of polymyxins for the management of multidrug-resistant gramnegative bacterial infections[END_REF], as well as the use of combinations of antibiotic agents for the treatment of patients with infections may be useful "weapons" in the "battle" against antimicrobial resistance.

Fosfomycin is an old antibiotic agent that has been used for the treatment of uncomplicated urinary tract infections in many clinical settings [START_REF] Knottnerus | Fosfomycin tromethamine as second agent for the treatment of acute, uncomplicated urinary tract infections in adult female patients in the Netherlands?[END_REF][START_REF] Patel | Fosfomycin tromethamine. A review of its antibacterial activity, pharmacokinetic properties and therapeutic efficacy as a single-dose oral treatment for acute uncomplicated lower urinary tract infections[END_REF]. Additionally, data reported from recent studies encourage the use of fosfomycin for the treatment of infections beyond the urinary tract [START_REF] Falagas | Fosfomycin: use beyond urinary tract and gastrointestinal infections[END_REF][START_REF] Falagas | Fosfomycin for the treatment of infections caused by multidrug-resistant non-fermenting Gram-negative bacilli: a systematic review of microbiological, animal and clinical studies[END_REF] as well as for infections due to specific categories of resistant bacteria [START_REF] Prakash | Oral and parenteral therapeutic options for outpatient urinary infections caused by enterobacteriaceae producing CTX-M extended-spectrum beta-lactamases[END_REF][START_REF] Rodriguez-Bano | Community infections caused by extended-spectrum beta-lactamase-producing Escherichia coli[END_REF]. Consequently, a potential synergistic activity of fosfomycin with other antibiotic agents may prove to be an important issue with considerable clinical implications.

In this regard, we aimed to collect and evaluate the available evidence regarding the in vitro synergistic activity of fosfomycin in combination with other antibiotic agents against Gram-negative and Gram-positive bacteria.

Methods

Literature search

We searched the PubMed and the Cochrane Library databases. The search strategy applied to these databases was "(fosfomycin OR phosphomycin OR phosphonomycin) AND (synergy OR synergism OR combination OR antagonism)." Bibliographies of relevant articles were also hand-searched.

Study selection criteria

A study was considered as eligible for inclusion if it provided data regarding the in vitro use of fosfomycin in combination with other antibiotic agents against Grampositive and/or Gram-negative microbial strains. Studies written in languages other than English, Spanish, French, German, Italian, or Dutch were excluded. Abstracts from scientific conferences were also excluded.

Data extraction

We extracted data regarding the laboratory methods used, the fosfomycin break point, the susceptibility of the pathogens to fosfomycin, the pathogens isolated, the site of isolation, and the type of antibiotics used in the combined therapy. Data regarding the number of bacterial strains for which synergism between fosfomycin and other selected antibiotic agents was observed were also extracted.

Results

Selected studies

Our searches performed in PubMed and the Cochrane Library generated a total of 411 and 36 articles, respectively. Among these, 41 studies were considered eligible for inclusion in our review .

Characteristics of the included studies

Twenty-seven of the 41 included studies provided data regarding the number of Gram-positive strains for which a synergistic effect of fosfomycin with other selected antibiotic agents was exhibited [10, 13, 15, 19, 21, 23-26, 28-33, 35-44, 46, 47]. Fifteen of the 41 included studies provided data regarding the number of Gram-negative strains for which a synergistic effect of fosfomycin with other selected antibiotic agents was exhibited [9, 11, 12, 14, 16-18, 20, 22, 27, 34, 45, 47-49]. One of the 41 included studies provided data regarding the number of Gram-positive as well as Gram-negative strains for which a synergistic effect of fosfomycin with other selected antibiotic agents was exhibited [START_REF] Olay | Interaction of fosfomycin with other antimicrobial agents: in vitro and in vivo studies[END_REF].

The definition of synergy used by the majority of the included studies (25 of the 41 included) was the fractional inhibitory concentration index (FICI), represented by the formula FICI=(MIC of fosfomycin in combination/MIC of fosfomycin alone)+(MIC of the antibiotic in combination/ MIC of the antibiotic alone), where MIC is the minimal inhibitory concentration. The efficacy time index (ETI) technique was also used in two of the included studies for the assessment of the synergistic effect of fosfomycin with the evaluated antibiotics [START_REF] Nakazawa | Enhancement of antimicrobial effects of various antibiotics against methicillin-resistant Staphylococcus aureus (MRSA) by combination with fosfomycin[END_REF][START_REF] Okazaki | Effectiveness of fosfomycin combined with other antimicrobial agents against multidrug-resistant Pseudomonas aeruginosa isolates using the efficacy time index assay[END_REF]. Detailed data regarding the laboratory methods used, the fosfomycin break points, the definition of synergy used, the antibiotics evaluated, as well as the pathogens evaluated along with the site of isolation are presented, when available, in Table 1.

Synergy of fosfomycin with other antibiotics

Gram-positive strains

Thirteen studies provided data regarding the in vitro synergistic effect of fosfomycin with other antibiotics against MRSA strains [13, 19, 26, 29-31, 36, 41-44, 46]. The study examining the largest number of MRSA strains reported a good synergistic effect of fosfomycin with cefamandole and with methicillin (65.5 and 31% of the 148 tested strains, respectively) [START_REF] Alvarez | In vitro activity of fosfomycin, alone and in combination, against methicillin-resistant Staphylococcus aureus[END_REF]. However, the findings of this study were not encouraging regarding the potential synergistic effect of fosfomycin with gentamicin, trimethoprim, and vancomycin [START_REF] Alvarez | In vitro activity of fosfomycin, alone and in combination, against methicillin-resistant Staphylococcus aureus[END_REF]. Two other studies providing data for a considerable number of MRSA strains reported a synergistic effect of 100% for the combination of fosfomycin with cefamandole [START_REF] Fosse | In vitro study of the cefamandole-fosfomycin combination against methicillin-resistant staphylococci[END_REF], and a synergistic effect of 94.2% for the combination of fosfomycin with cefazolin [START_REF] Chang | Effects of antibiotic combinations on methicillin-resistant Staphylococcus aureus in vitro[END_REF]. Encouraging relevant data were also reported from several other studies, although the number of the evaluated MRSA was rather limited. Specific data are shown in Table 2.

Nine studies provided data regarding the in vitro synergistic effect of fosfomycin with other antibiotics against Staphylococcus aureus and/or other coagulasenegative Staphylococcus strains [START_REF] Jm | In vitro activity of fosfomycin against 'problem' gram-positive cocci[END_REF][START_REF] Gatermann | The microbiological efficacy of the combination of fosfomycin and vancomycin against clinically relevant staphylococci[END_REF][START_REF] Courcol | In-vitro activity of the combination of ceftriaxone and fosfomycin against staphylococci[END_REF][START_REF] David | In vitro study of a ceftriaxonefosfomycin combination on 55 strains of methicillin-sensitive and heterogeneous methicillin-resistant Staphylococcus aureus[END_REF][START_REF] May | Treatment of post-traumatic and post-neurosurgical bacterial meningitis with ceftriaxone alone or in combination with fosfomycin[END_REF][START_REF] Ullmann | Synergism between ciprofloxacin and fosfomycin in vitro[END_REF][START_REF] Debbia | In vitro activity of imipenem against enterococci and staphylococci and evidence for high rates of synergism with teicoplanin, fosfomycin, and rifampin[END_REF][START_REF] Stahl | Treatment of neurosurgical bacterial meningitis using the combination of ceftriaxonefosfomycin[END_REF][START_REF] Olay | Interaction of fosfomycin with other antimicrobial agents: in vitro and in vivo studies[END_REF]; specific data are shown in Table 2. Notably, a synergistic effect of fosfomycin with ciprofloxacin was reported for 51.3% of the 37S. aureus tested strains [START_REF] Ullmann | Synergism between ciprofloxacin and fosfomycin in vitro[END_REF]. The findings regarding gentamicin were discouraging (0 of the 29S. aureus tested strains) [START_REF] Olay | Interaction of fosfomycin with other antimicrobial agents: in vitro and in vivo studies[END_REF]. Relevant data regarding Streptococcus spp. were reported from five studies [START_REF] Totsuka | In vitro combined effects of fosfomycin and β-lactam antibiotics against penicillin-resistant Streptococcus pneumoniae[END_REF][START_REF] Kikuchi | Effects of combination of benzylpenicillin and fosfomycin on penicillin-resistant Streptococcus pneumoniae[END_REF][START_REF] Doit | In vitro killing activities of antibiotics at clinically achievable concentrations in cerebrospinal fluid against penicillin-resistant Streptococcus pneumoniae isolated from children with meningitis[END_REF][START_REF] Barakett | Bacteriostatic activity and killing curves of eight antibiotics against seven strains of penicillin G-resistant pneumococci[END_REF][START_REF] May | Treatment of post-traumatic and post-neurosurgical bacterial meningitis with ceftriaxone alone or in combination with fosfomycin[END_REF]. A similar synergistic effect of fosfomycin was reported with penicillin [START_REF] Totsuka | In vitro combined effects of fosfomycin and β-lactam antibiotics against penicillin-resistant Streptococcus pneumoniae[END_REF], cefminox [START_REF] Totsuka | In vitro combined effects of fosfomycin and β-lactam antibiotics against penicillin-resistant Streptococcus pneumoniae[END_REF], cefotaxime [START_REF] Barakett | Bacteriostatic activity and killing curves of eight antibiotics against seven strains of penicillin G-resistant pneumococci[END_REF], and cefotiam [START_REF] Totsuka | In vitro combined effects of fosfomycin and β-lactam antibiotics against penicillin-resistant Streptococcus pneumoniae[END_REF]. Specific data are shown in Table 2.

Three studies provided data regarding the in vitro synergistic effect of fosfomycin with other antibiotics against Enterococcus spp. [START_REF] Pestel | In vitro interactions between different beta-lactam antibiotics and fosfomycin against bloodstream isolates of enterococci[END_REF][START_REF] Rice | In vitro synergism between daptomycin and fosfomycin against Enterococcus faecalis isolates with high-level gentamicin resistance[END_REF][START_REF] Debbia | In vitro activity of imipenem against enterococci and staphylococci and evidence for high rates of synergism with teicoplanin, fosfomycin, and rifampin[END_REF]. A synergistic effect of 90% was reported for the combination of fosfomycin with cefotaxime from a study analyzing a total of 50 strains [START_REF] Pestel | In vitro interactions between different beta-lactam antibiotics and fosfomycin against bloodstream isolates of enterococci[END_REF]. Data regarding the combination of fosfomycin with daptomycin [START_REF] Rice | In vitro synergism between daptomycin and fosfomycin against Enterococcus faecalis isolates with high-level gentamicin resistance[END_REF] or imipenem [START_REF] Debbia | In vitro activity of imipenem against enterococci and staphylococci and evidence for high rates of synergism with teicoplanin, fosfomycin, and rifampin[END_REF] were also encouraging.

Gram-negative strains

Thirteen studies provided data regarding the in vitro synergistic effect of fosfomycin with other antibiotics against Pseudomonas aeruginosa strains [9, 11, 12, 14, 16-18, 20, 27, 34, 38, 45, 47]. In one of the abovementioned studies providing data for 30 P. aeruginosa strains [START_REF] Okazaki | Effectiveness of fosfomycin combined with other antimicrobial agents against multidrug-resistant Pseudomonas aeruginosa isolates using the efficacy time index assay[END_REF], fosfomycin exhibited a synergistic effect in≥ 70% of the tested strains when it was used in combination with aztreonam, cefepime, ceftazidime, gentamicin, imipenem, and levofloxacin. Two other studies evaluated the potential synergistic effect of fosfomycin with ciprofloxacin, reporting data for 74 and 37 P. aeruginosa strains, respectively; the observed synergistic effect was 27% [START_REF] Yamada | Role of ciprofloxacin in its synergistic effect with fosfomycin on drugresistant strains of Pseudomonas aeruginosa[END_REF] and 78.4% [START_REF] Ullmann | Synergism between ciprofloxacin and fosfomycin in vitro[END_REF], respectively. Three other studies evaluated the potential synergistic effect of fosfomycin with amikacin against P. aeruginosa [START_REF] Traub | Surveillance of Pseudomonas aeruginosa in intensive care units: clusters of nosocomial cross-infection and encounter of a multiple-antibiotic resistant strain[END_REF][START_REF] Tessier | In vitro activity of fosfomycin combined with ceftazidime, imipenem, amikacin, and ciprofloxacin against Pseudomonas aeruginosa[END_REF][START_REF] Watine | Susceptibility of multiresistant serotype 012 Pseudomonas aeruginosa to fosfomycin in combination with other antibiotics[END_REF]. The observed synergistic effect ranged from 18.8 to 100%. Yet, the number of the analyzed strains in each of these four studies was limited (maximum 26 strains).

A single study provided data regarding the potential synergistic effect of fosfomycin against a total of 34 Acinetobacter baumannii strains [START_REF] Martinez-Martinez | In-vitro activity of antimicrobial agent combinations against multiresistant Acinetobacter baumannii[END_REF]. Specific data are shown in Table 3. Another study provided data regarding the synergistic effect of fosfomycin with gentamicin against Escherichia coli and Klebsiella spp. [START_REF] Olay | Interaction of fosfomycin with other antimicrobial agents: in vitro and in vivo studies[END_REF]. Specific data are shown in Table 3. Yet, the number of the evaluated strains was limited-30 and 12 strains, respectively. Data regarding Salmonella and Shigella strains were also reported in this study [START_REF] Olay | Interaction of fosfomycin with other antimicrobial agents: in vitro and in vivo studies[END_REF]. A synergistic effect for the combination of fosfomycin with amikacin was observed in 82.2% of the 90 evaluated Salmonella strains. Data regarding the synergistic effect of fosfomycin with carbenicillin and gentamicin against Serratia marcescens were provided in two studies [START_REF] Olay | Interaction of fosfomycin with other antimicrobial agents: in vitro and in vivo studies[END_REF][START_REF] Baquero | Antibiotherapy of Serratia marcescens septicemia in children[END_REF]. The findings were discordant; specific data are shown in Table 3.

Discussion

The findings of our study imply that the combination of fosfomycin with other antibiotic agents may provide a useful alternative regarding the treatment of patients with Gram-negative or Gram-positive infections, including MRSA infection. However, the evaluated in vitro data, generated mainly from old studies, do not prove directly the in vitro synergism of fosfomycin with other antibiotic agents against the above-mentioned infections. Additionally, the considerable heterogeneity among the included studies regarding the definition of synergy and the methodology used hampers the establishment of firm conclusions, as well as the extrapolation of the in vitro findings into clinical practice.

Fosfomycin is an antibiotic agent that has been mainly used for the treatment of urinary tract infections. Specifically, fosfomycin tromethamine is approved as an oral single-dose treatment for acute uncomplicated cystitis [START_REF] Warren | Guidelines for antimicrobial treatment of uncomplicated acute bacterial cystitis and acute pyelonephritis in women. Infectious Diseases Society of America (IDSA)[END_REF]. Moreover, fosfomycin may also be a useful treatment option for specific types of resistant uropathogens that exhibit considerably high rates of resistance due to the production of extended spectrum beta-lactamases (ESBLs) [START_REF] Hernandez | In vitro activity of fosfomycin against ESBL-producing enterobacteria of urinary origin[END_REF][START_REF] Zahar | Addressing the challenge of extended-spectrum betalactamases[END_REF]. However, recent studies suggest that mutations occurring among populations of uropathogens and specifically among populations of ESBL producers may result in the emergence of resistance to fosfomycin [START_REF] Ellington | Mutators among CTX-M beta-lactamase-producing Escherichia coli and risk for the emergence of fosfomycin resistance[END_REF][START_REF] Horii | Emergence of fosfomycin-resistant isolates of Shiga-like toxinproducing Escherichia coli O26[END_REF]. Indeed, the issue of emergence of resistance is one of significant importance regarding fosfomycin treatment. Resistance of fosfomycin may be attributed to several chromosomal mutations, which mainly affect its uptake [START_REF] Nilsson | Biological costs and mechanisms of fosfomycin resistance in Escherichia coli[END_REF]. As a result, the potential of a synergistic effect of fosfomycin with another antibiotic agent is an issue of importance, both at a patient and at a community level.

According to a recently published review, pharmacokinetic and pharmacodynamic aspects of fosfomycin, such as the limited binding with serum proteins, the unchanged excretion in the urine, along with the relatively high concentrations that it achieves in the urine, serum, lung, cerebrospinal fluid, and other sites, favor its use for the treatment of relevant infections [START_REF] Roussos | Clinical significance of the pharmacokinetic and pharmacodynamic characteristics of fosfomycin for the treatment of patients with systemic infections[END_REF]. Regarding the potential synergistic effect of fosfomycin with other antibiotic agents, fosfomycin appears to have potentially useful pharmacokinetic/pharmacodynamic aspects that favor its use in combination with other antibiotic agents for the treatment of specific infections. In brief, the ability of n/N Number of isolates exhibiting synergism between fosfomycin and the examined antimicrobial agent/total number of isolates examined in the included study a In cases where fewer than 10 bacterial strains were evaluated, percentages are not displayed in the table fosfomycin to modify the production of penicillin-binding proteins (PBPs) enables its use for the treatment of MRSA infections as well as infections from penicillin-resistant S. pneumoniae [START_REF] Kikuchi | Effects of combination of benzylpenicillin and fosfomycin on penicillin-resistant Streptococcus pneumoniae[END_REF][START_REF] Utsui | Antibacterial activity of cefmetazole alone and in combination with fosfomycin against methicillin-and cephem-resistant Staphylococcus aureus[END_REF]. The ability of fosfomycin to penetrate biofilm layers encourages the use of fosfomycin in combination with other antibiotics against microbial strains growing on biofilms. Specifically, relevant studies report a synergistic effect of fosfomycin with quinolone antibiotic agents such as ulifloxacin, levofloxacin [START_REF] Mikuniya | Synergistic effect of fosfomycin and fluoroquinolones against Pseudomonas aeruginosa growing in a biofilm[END_REF], and ofloxacin against P. aeruginosa biofilm isolates [START_REF] Monden | Role of fosfomycin in a synergistic combination with ofloxacin against Pseudomonas aeruginosa growing in a biofilm[END_REF]. Notably, fosfomycin is synergistic with N-acetylocysteine against E. coli biofilm isolates [START_REF] Marchese | Effect of fosfomycin alone and in combination with N-acetylcysteine on E. coli biofilms[END_REF].

Another aspect of fosfomycin that could prove useful, when it is used in combination with other antibiotic agents, is its ability to modify the toxicity of many types of co-administered drugs [START_REF] Roussos | Clinical significance of the pharmacokinetic and pharmacodynamic characteristics of fosfomycin for the treatment of patients with systemic infections[END_REF]. Specifically, fosfomycin is reported to mitigate in vivo the toxicity of aminoglycosides [START_REF] Inouye | Mode of protective action of fosfomycin against dibekacin-induced nephrotoxicity in the dehydrated rats[END_REF], glycopeptides [START_REF] Yoshiyama | The effect of fosfomycin on glycopeptide antibioticinduced nephrotoxicity in rats[END_REF], as well as polymyxin B [START_REF] Leach | Effect of topical fosfomycin on polymyxin B ototoxicity[END_REF]. Finally, it has been reported that the synergistic combination with either amoxicillin or clarithromycin may provide a useful alternative treatment option for patients with Helicobacter pylori infections [START_REF] Blacky | In vitro activity of fosfomycin alone and in combination with amoxicillin, clarithromycin and metronidazole against Helicobacter pylori compared with combined clarithromycin and metronidazole[END_REF].

Our study has limitations that should be taken into consideration before the extrapolation of our findings. Firstly, considerable heterogeneity is observed among the included studies regarding the definition of synergy; specifically only 25 of the 41 studies used the fractional inhibitory concentration index as the definition of synergy of fosfomycin with the respective evaluated antibiotics. Heterogeneity was also observed regarding the laboratory n/N Number of isolates exhibiting synergism between fosfomycin and the examined antimicrobial agent/total number of isolates examined in the included study. a In cases where fewer than 10 bacterial strains were evaluated, percentages are not displayed in the table methods, as well as the fosfomycin break points used. Moreover, the majority (34 of the 41) of the included studies were older studies, published during the period 1977-1999. In this regard, we were unable to reach conclusive results regarding specific antibiotic combinations with fosfomycin for the evaluated microorganisms. Despite these shortcomings, the available data are not to be overlooked as they indicate that further research is justified.

In conclusion, in this era of alarmingly high drug resistance, the synergistic combination of fosfomycin with several other antibiotic agents may be a useful alternative treatment option for Gram-negative and Gram-positive infections. However, additional studies with more stringent definitions, as well as studies that will evaluate the clinical efficacy of synergistic combinations of fosfomycin with other antibiotics will contribute to the already available knowledge regarding the role of fosfomycin in synergistic combinations with other antibiotics.

Funding None

Conflict of interest None

Table 1

 1 Main characteristics of the studies included in the review

	First author,	Laboratory method	Fosfomycin	Definition of synergy	Antibiotics	Pathogens evaluated/site
	year [ref]		break point		evaluated	of isolation
	Yamada et	Broth microdilution	CLSI criteria	FICI≤0.5	Ciprofloxacin	Ciprofloxacin-resistant P.
	al., 2007					aeruginosa/NR
	[9]
	Sahuquillo	Checkerboard method	Lowest	FICI≤0.5	Linezolid	MSSA/NR
	Arce et		concentration
	al., 2006		that
	[10]		prevented
			growth after
			24 h, NR
	Mikuniya et	Checkerboard method	NR	FICI≤0.5	Ulifloxacin	P. aeruginosa/NR
	al., 2005				Levofloxacin
	[11]
	Pruekprasert	Checkerboard titration	CLSI criteria	FICI≤0.5	Gentamicin	P. aeruginosa/NR
	et al.,	method			Ceftazidime
	2005 [12]				Imipenem
	Nakazawa	Broth microdilution	NR	ETI assay 1-8 index=synergy Imipenem	MRSA/wounds of burn patients
	et al.,	method designed by			Cefmetazole
	2003 [13]	the Japanese Society of Chemotherapy			Sulbactam/ cefoperazone
					Panipenem
					Ampicillin
					Cefotiam
					Flomoxef
					sodium
					Arbekacin
					Vancomycin
					Minocycline
					Ofloxacin
	Okazaki et	Efficiency time index	≤16	ETI ≥1 to <8 good synergism,	Cefepime	P. aeruginosa/NR
	al., 2002	assay		≥ 8 excellent synergism	Ceftazidime
	[14]				Aztreonam
					Imipenem
					Meropenem
					Gentamicin
					Piperacillin
					Levofloxacin
	Grif et al.,	Checkerboard method	≤16	NR	Rifampicin	S. epidermidis, S. aureus/catheter
	2001 [15]	and killing curve			Linezolid	tip
		method			Quinupristin/
					dalfopristin
					Moxifloxacin
					Vancomycin
					Cefazolin
					Meropenem
	Hayami et	Checkerboard method	NR	FICI≤0.5	Meropenem	P. aeruginosa/UTI
	al., 1999	and killing curve			Amikacin
	[16]	method, agar dilution			Ceftazidime
					Ciprofloxacin
	Traub et al.,	Time kill	CLSI criteria	NR	Amikacin	MDR P. aeruginosa/trachea
	1998 [17]
	Dubrous et	Agar dilution	≤16	NR	Ceftazidime	P. aeruginosa/NR
	al., 1997
	[18]

Table 1 (

 1 continued)

	First author,	Laboratory method	Fosfomycin	Definition of synergy	Antibiotics	Pathogens evaluated/site
	year [ref]		break point		evaluated	of isolation
	Gatermann	Agar dilution	≤16	FICI≤0.5	Vancomycin	S. aureus, S. epidermidis/
	et al.,					abscesses, wound infections,
	1989 [32]					endocarditis, foreign body
						infections, UTI
	Rice et al.,	Agar dilution, time kill	64	Time-kill techniques	Daptomycin	High-level gentamicin resistance,
	1989 [33]	experiments				Enterococcus faecalis/NR
	Figuerdo et	Agar plate dilution	NR	FICI≤0.5	Ciprofloxacin	P. aeruginosa/sputum
	al., 1988	checkerboard method			Ofloxacin
	[34]
	Courcol et	Checkerboard method	≤25	FICI≤0.5	Ceftriaxone	S. aureus (3 MRSA), S. epidermitis
	al., 1987	and killing curve				(3 MRSE); all strains beta
	[35]	method, broth				lactamase producing/blood
		microdilution for the
		MIC combinations
	David et al.,	Checkerboard method	NR	FIC≤0.5	Ceftriaxone	MRSA, MSSA/NR
	1987 [36]
	May et al.,	Microbroth dilution	NR	FICI≤0.5	Ceftriaxone	S. aureus, S. pneumoniae,
	1987 [37]					Klebsiella spp., Peptococcus/CSF
	Ullmann et	Checkerboard	NR	FICI≤0.5	Ciprofloxacin	P. aeruginosa, S. aureus/NR
	al., 1987
	[38]
	Debbia et	Checkerboard method	R ≥ 32	FICI≤0.5	Imipenem	E. faecalis, E. faecium, S. aureus
	al., 1986	and killing curve				(16 MRSA), S. epidermidis (7
	[39]	method				MRSE), Staphylococcus spp./NR
	Stahl et al.,	Microdiluition	NR	FICI≤0.5	Ceftriaxone	S. epidermidis, S. aureus, Neisseria
	1986 [40]					meningitidis, S. pneumoniae,
						Haemophilus influenzae, Serratia
						marcescens, Aeromonas
						hydrophila/CSF
	Utsui et al.,	Checkerboard	NR	FICI≤0.5	Cefmetazole	MRSA/NR
	1986 [41]				Cefaloridine
					Cefotaxime
	Alvarez et	Agar dilution	≤16	MICs of both drugs decreased	Vancomycin	MRSA/blood, sputum, wounds,
	al., 1985			by at least 1/4	Gentamicin	urine, peritoneal fluid, and eyes
	[42]				Cefamandole
					Methicillin
					Trimethoprim
	Portier et	Checkerboard method	NR	FBC≤0.75	Cefotaxime	MRSA also resistant to gentamicin/
	al., 1985				Cefoperazone	CSF, bone and joint infection,
	[43]				Cefamandole	thrombophlebitis
					Cephalothin
	Fosse et al.,	Checkerboard method,	≤ 8	FICI≤0.5	Cefamandole	MR Staphylococcus spp., MRSA,
	1984 [44]	broth microdilution			Vancomycin	MRSE/NR
					Rifampicin
	Takahashi et	Agar plate dilution	NR	FICI≤0.5	Piperacillin-	P. aeruginosa/NR
	al., 1984	checkerboard method			tobramycin
	[45]				Piperacillin
					Cefoperazone
					Cefoperazone-
					tobramycin
					Cefsulodin
					Cefsulodin-
					tobramycin

Table 1

 1 (continued) NR Not reported, R resistance, MIC minimum inhibitory concentration, FIC fractional inhibitory concentrations, FICI fractional inhibitory concentration index, MRSA methicillin-resistant Staphylococcus aureus, MSSA methicillin-susceptible Staphylococcus aureus, CSF cerebrospinal fluid, MRSE methicillin-resistant Staphylococcus epidermidis, CFU colony forming units, CLSI Clinical and Laboratory Standards Institute, UTI urinary tract infections, ETI efficacy time index, FBC fractional bactericidal concentration

	First author,	Laboratory method	Fosfomycin	Definition of synergy	Antibiotics	Pathogens evaluated/site
	year [ref]		break point		evaluated	of isolation
	Tremeux et	Checkerboard	NR	FICI≤0.5	Cefotaxime	MRSA/CSF
	al., 1983
	[46]
	Olay et al.,	Agar plate checkerboard	NR	4 times or greater reduction of	Gentamicin	P. aeruginosa, S. aureus,
	1978 [47]	method		the MIC of both antibiotics	Carbenicillin	Streptococcus spp., E. coli,
				compared to the MIC of each		Klebsiella spp., Proteus spp.,
				antibiotic alone		Entrococcus cloacae, Salmonella
						spp., Serratia marcescens/NR
	Perea et al.,	Agar dilution method,	NR	FICI≤0.75	Ampicillin	Salmonella spp., Shigella spp./NR
	1978 [48]	checker board titration			Chloramphenicol
	Baquero et	Agar dilution,	≤32	FICI≤0.5	Gentamicin	Serratia marscesens/NR
	al., 1977	checkerboard method			Carbenicillin
	[49]

Table 2

 2 Data retrieved from the studies examining the synergistic effect between fosfomycin and specific other antimicrobial agents for Gram-positive bacterial strains a

	Antibiotics	n/N (%) [ref]
	MRSA
	Ampicillin	7/32 (21.8) [13]
	Cefamandole	97/148 (65.5) [42], 10/10 (100) [43], 50/50
		(100) [44]
	Cefazolin	66/70 (94.2) [31], 11/25 (44) [30]
	Cefmetazole	26/32 (81.2) [13], 45/70 (64.2) [31], 10/14
		(71.4) [41]
	Cefotaxime	9/14 (64.2) [41], 10/10 (100) [43], 1/1 [46]
	Cefotiam	7/32 (21.8) [13]
	Cefperazone	10/10 (100) [43]
	Ceftriaxone	20/30 (66.6) [36]
	Ciprofloxacin	14/18 (77.7) [29]
	Fusidic acid	0/24 (0) [26]
	Gentamicin	10/148 (6.7) [42]
	Imipenem	16/32 (50) [13]
	Linezolid	4/5 [10]
	Methicillin	46/148 (31) [42]
	Oxacillin	3/16 (18.7) [19]
	Rifampicin	15/19 (78.9) [29]. 0/50 (0) [44]
	Sparfloxacin	0/16 (0) [19]
	Sulbactam/	19/32 (59.3) [13]
	cefoperazone
	Trimethoprim	4/148 (2.7) [42]
	Vancomycin	32/32 (100) [13], 0/148 (0) [42], 10/50 (20)
		[44]
	Staphylococcus aureus
	Ceftriaxone	2/6 [35], 11/23 (47.8) [36], 0/3 [37], 0/2 [40]
	Ciprofloxacin	19/37 (51.3) [38]
	Imipenem	10/20 (50) [39]
	Vancomycin	1/15 (6.6) [32], 0/5 [15]
	Gentamicin	0/29 (0) [47]
	Rifampicin	13/38 (34.8) [47], 5/5 [15]
	Coagulase-negative staphylococci
	Ciprofloxacin	12/19 (63.1) [29]
	Imipenem	10/10 (100) [39]
	Rifampin	7/15 (46.6) [29]
	Vancomycin	1/18 (5.5) [32]
	Streptococcus spp.
	Penicillin	17/56 (30.4) [21], 9/51 (17.6) [23]
	Cefazolin	3/56 (5.3) [21]
	Cefepime	0/56 (0) [21]
	Cefminox	33/56 (58.9) [21]
	Cefotaxime	3/56 (5.4) [21], 3/7 (42.9) [28]
	Cefotiam	20/56 (35.7) [21]
	Ceftriaxone	0/26 (0) [25], 0/4 [37]
	Imipenem	1/56 (1.8) [21], 0/26 (0) [25]
	Vancomycin	0/26 (0) [28]

Table 2

 2

	(continued)
	Antibiotics	n/N (%) [ref]
	Enterococcus spp.
	Cefotaxime	45/50 (90.0) [24]
	Daptomycin	20/64 (31.3) [33]
	Imipenem	19/36 (52.7) [39]

Table 3

 3 Data retrieved from the studies examining the synergistic effect between fosfomycin and specific other antimicrobial agents for Gramnegative bacterial strains a

	Antibiotics	n/N (%) [ref]
	Non-fermentative Gram-negative strains
	Pseudomonas aeruginosa
	Amikacin	0/26 (0) [16], 2/2 (100) [17], 3/16 (18.8) [20], 8/12 (66.7) [27]
	Aztreonam	23/30 (76.7) [14], 7/12 (58.3) [27]
	Cefepime	23/30 (76.7) [14]
	Ceftazidime	2/18 (11.1) [12], 21/30 (70.0) [14], 7/26 (26.9) [16], 8/12 (66.7) [18], 0/16 (0) [20], 9/12 (75) [27]
	Ciprofloxacin	20/74 (27.0) [9], 10/26 (38.4) [16], 6/16 (37.5) [20], 29/37 (78.4) [38], 18/30 (60) [34]
	Gentamicin	1/22 (4.5) [12], 21/30 (70.0) [14], 55/77 (71.4) [47]
	Imipenem	11/29 (37.9) [12], 22/30 (73.3) [14], 0/16 (0) [20], 4/12 (33.3) [27]
	Levofloxacin	1/1 [11], 20/30 (86.9) [14]
	Acinetobacter baumannii
	Amikacin	15/34 (38.2) [22]
	Ceftazidime	1/34 (2.9) [22]
	Ciprofloxacin	1/34 (2.9) [22]
	Imipenem	1/34 (2.9) [22]
	Enterobacteriaceae
	Escherichia coli
	Gentamicin	4/30 (13.4) [47]
	Klebsiella pneumoniae
	Gentamicin	5/12 (41.7) [47]
	Salmonella spp.
	Amikacin	74/90 (82.2) [48]
	Cefepime	56/90 (62.2) [48]
	Serratia marcescens
	Carbenicillin	21/24 (87.5) [49], 9/32 (28.1) [47]
	Gentamicin	22/24 (91.6) [49], 5/39 (12.8) [47]
	Shigella spp.
	Amikacin	27/50 (48) [48]
	Cefepime	29/50 (58) [48]


J W A J W Huijsdens

Kluytmans

E J M Verkade
email: everkade@amphia.nl

C J M M Verhulst

X W Huijsdens

J A J W Kluytmans

In vitro activity of tigecycline against methicillin-resistant Staphylococcus aureus, including livestock-associated strains

In vitro activity of tigecycline against methicillin-resistant , including livestock-associated strains

Introduction

Infections caused by methicillin-resistant Staphylococcus aureus (MRSA) have traditionally been a problem in healthcare settings [START_REF] Tiemersma | European Antimicrobial Resistance Surveillance System Participants (2004) Methicillin-resistant Staphylococcus aureus in Europe, 1999-2002[END_REF]. According to a report from the National Nosocomial Infections Surveillance (NNIS) System, approximately 60% of all S. aureus isolated from patients in intensive care units in US hospitals were methicillinresistant in 2003 [START_REF]National Nosocomial Infections Surveillance (NNIS) System Report, data summary from January 1992 through June 2004, issued[END_REF]. For the last approximately 10 years, MRSA has expanded its territory to the community, causing severe infections in previously healthy persons all over the world [START_REF] Diederen | The emergence of infections with community-associated methicillin resistant Staphylococcus aureus[END_REF][START_REF] Kluytmans-Vandenbergh | Communityacquired methicillin-resistant Staphylococcus aureus: current perspectives[END_REF]. In 2003, a new clone of MRSA was observed in The Netherlands that is related to an extensive reservoir in pigs and cattle [START_REF] Voss | Methicillin-resistant Staphylococcus aureus in pig farming[END_REF]. The livestock-associated clone is characterized by being non-typable by SmaI pulsed-field gel electrophoresis (PFGE). By the end of 2007, nearly 30% of all MRSA observed in The Netherlands were of this type [START_REF] Huijsdens | The clonal structure of PFGE non-typeable methicillinresistant Staphylococcus aureus in the Netherlands[END_REF]. There are important differences between livestockassociated MRSA (LA-MRSA), healthcare-associated MRSA (HA-MRSA), and community-associated MRSA (CA-MRSA) regarding the susceptibility against antimicrobial agents. HA-MRSA isolates are frequently multidrugresistant, while CA-MRSA and LA-MRSA are relatively susceptible for most non-beta-lactam antibiotics, with the exception of tetracycline for LA-MRSA, for which they are almost always resistant. This is most likely due to the extensive use of this antimicrobial agent in animal husbandry. Because tigecycline is related to tetracycline, it is important to determine the activity of this new drug for LA-MRSA.

The treatment of serious MRSA infections has been based, for many years, upon the use of glycopeptides, i.e., vancomycin and teicoplanin. However, concerns over increasing rates of heteroresistance and tolerance to glycopeptides [START_REF] Chang | Vancomycin-Resistant Staphylococcus aureus Investigative Team (2003) Infection with vancomycin-resistant Staphylococcus aureus containing the vanA resistance gene[END_REF] has urged the development of newer agents. Tigecycline is the first commercially available member of the glycylcyclines, a new class of antimicrobial agents. The glycylcyclines are derivatives of the tetracycline antibiotics, with structural modifications that result in activity against gram-positive, gram-negative, and anaerobic micro-organisms, including multidrug-resistant strains. It exhibits generally bacteriostatic action by reversibly binding to the 30S ribosomal subunit and inhibiting protein translation [START_REF] Rose | Tigecycline: first of a new class of antimicrobial agents[END_REF].

The purpose of the present study was to assess the in vitro activity of tigecycline against MRSA isolates collected in The Netherlands using a well-defined collection of strains that included a representative sample of LA-MRSA strains.

Materials and methods

A total of 202 MRSA isolates were tested in this study. All MRSA isolates are part of the MRSA strain collection of the National Institute of Public Health and Environmental Protection (RIVM), Bilthoven, The Netherlands. The collection consisted of three subsets. The first set of isolates used in this study contained 76 MRSA isolates that were collected between 1990 and 1998 in The Netherlands (old MRSA). The second set was 93 MRSA isolates collected between 2003 and 2005 (recent MRSA). These MRSA strains all had a unique PFGE typing result. The third set of isolates tested consisted of 33 LA-MRSA strains and were collected between 2003 and 2005. They had been collected in a previous study and the strains in our evaluation are the index cases of the previous survey [START_REF] Van Loo | Emergence of methicillin-resistant Staphylococcus aureus of animal origin in humans[END_REF]. All 202 isolates have been confirmed as S. aureus and methicillin-resistant using a duplex polymerase chain reaction (PCR) for the mecA gene and coagulase gene as described previously [START_REF] Kluytmans | Performance of CHROMagar selective medium and oxacillin resistance screening agar base for identifying Staphylococcus aureus and detecting methicillin resistance[END_REF][START_REF] Van Griethuysen | Rapid slide latex agglutination test for detection of methicillin resistance in Staphylococcus aureus[END_REF].

The minimum inhibitory concentration (MIC) for tigecycline was determined by using the Etest system (AB Biodisk, Solna, Sweden) with a concentration range of 0.016 to 256 μg/ml. Etest strips contained a concentration gradient of the antimicrobial agent with a standard amount of calcium throughout the strip. Etest strips were applied to the surface of 150-mm Mueller-Hinton agar plates. Plates were incubated at 35°C in ambient air for 24 h prior to reading the MIC results. In addition, the MICs of the following antimicrobial agents were determined simultaneously: oxacillin, gentamicin, cotrimoxazole, ciprofloxacin, erythromycin, clindamycin, rifampin, daptomycin, tetracycline, linezolid, vancomycin, and teicoplanin. All MICs were determined using the Etest system. For vancomycin and teicoplanin, the Etest strips were placed on brain heart infusion agar, using a high inoculum (2.0 McFarland) and an extended incubation time (48 h) to be able to detect hGISA isolates. Isolates were categorized as susceptible or resistant to an antimicrobial agent according to the breakpoints published by the Clinical and Laboratory Standards Institute (CLSI) [START_REF]Performance standards for antimicrobial susceptibility testing[END_REF]. The proposed breakpoint for tigecycline is greater than 0.5 mg/l for S. aureus (both methicillin-resistant and methicillin-susceptible strains). The 11 MRSA strains with the highest MIC for tigecycline on Mueller-Hinton agar plates were subsequently applied on 90-mm Iso-Sensitest agar plates (Oxoid Ltd.). The plates were then incubated at 35°C in ambient air for 24 h prior to reading the MIC results.

All results were entered into a database and further statistical analyses were performed using SPSS software. The MIC values for all tested antimicrobial agents of the different subsets of strains were compared using the Mann-Whitney U-test.

Results and discussion

The observed MIC range for tigecycline was 0.05 to 1.0 μg/ml, with MIC values at which 50 and 90% of the isolates tested are inhibited (MIC 50 and MIC 90 ) of 0.19 and 0.38 μg/ml, respectively. The MIC 50 and MIC 90 of LA-MRSA, old MRSA, and recent MRSA isolates for tigecycline and other antibiotics are outlined in Table 1. No significant difference was found in the portion of tigecycline resistance between recent MRSA and old MRSA. None of the LA-MRSA isolates were resistant for tigecycline. Three (2%) of the 169 tested MRSA isolates were resistant for tigecycline. Of the 76 old MRSA isolates, two (3%) isolates had MICs for tigecycline greater than 0.5 μg/ml and are, therefore, considered to be resistant for tigecycline. In addition, one isolate (1%) of the 93 recent MRSA strains had an MIC value for tigecycline greater than 0.5 μg/ml. The 11 MRSA strains with the highest MIC for tigecycline on Mueller-Hinton agar plates were retested using Etest strips that were applied on Iso-Sensitest agar plates (Oxoid Ltd.) and simultaneously on Mueller-Hinton agar plates. The MIC values for tigecycline of the 11 MRSA strains applied on Mueller-Hinton agar plates were comparable with the previously obtained MIC values. On Iso-Sensitest medium, these 11 MRSA strains had significantly 2-fold lower MIC values for tigecycline using linear regression analysis (p< 0.001). Figure 1 shows the shift towards lower MIC values for tigecycline when MRSA strains were applied on Iso-Sensitest medium. The MICs for tigecycline showed a significant correlation with those of tetracycline (r=0.518; p<0.001) and teicoplanin (r=0.325; p<0.001).

LA-MRSA was the most susceptible group of strains. They were only significantly more often resistant to tetracycline (Table 1). Old MRSA strains were more often resistant to most groups of antimicrobial agents, compared to recent MRSA. The only agent that was significantly (p< 0.001) more resistant in recent strains in comparison with old strains was vancomycin.

In a well-defined collection of MRSA, we found that a minority (1.5%) was resistant for tigecycline. All strains were isolated before tigecycline had been used in patients. The results of this study slightly differed from other data on European and North American antibiotic-resistant clinical isolates that were phenotypically characterized [13][START_REF] Mendes | Antimicrobial activity of tigecycline against community-acquired methicillinresistant Staphylococcus aureus isolates recovered from North American medical centers[END_REF][START_REF] Zhanel | Antimicrobial-resistant pathogens in intensive care units in Canada: results of the Canadian National Intensive Care Unit (CAN-ICU) Study, 2005-2006[END_REF]. In a recent study, the in vitro activity of tigecycline against 38 MRSA and the correlation of this activity with their resistance gene content were determined [START_REF] Borbone | Evaluation of the in vitro activity of tigecycline against multiresistant Gram-positive cocci containing tetracycline resistance determinants[END_REF]. Tigecycline demonstrated good activity against MRSA, with MIC 50 and MIC 90 values of 0.12 and 0.25 μg/ml, respectively. Overall, tigecycline showed an MIC range of 0.06 to 0.25 μg/ml. The tigecycline MICs determined in our study were slightly higher.

In another study, Fluit et al. [START_REF] Fluit | Presence of tetracycline resistance determinants and susceptibility to tigecycline and minocycline[END_REF] found the MIC range for tigecycline to be 0.06 to 2.0 μg/ml. For the 106 S. aureus isolates tested, two (2%) isolates had MIC values for tigecycline greater than 0.5 μg/ml and are, therefore, considered to be resistant. These findings are identical to our results. In our study, the MICs for tigecycline showed a significant correlation with the MICs for tetracycline. Fluit et al. found no relation between the presence of tetracycline resistance determinants tet(K) or tet(M) and the MICs for tigecycline observed for S. aureus, although tetracyclinesusceptible isolates were more often susceptible to tigecycline.

The possible correlation of the in vitro susceptibility of tigecycline and tetracycline prompted us to include LA-MRSA in the evaluation. These strains are known to have high levels of tetracycline resistance. Also, the current evaluation showed that 28 out of the 33 (85%) LA-MRSA strains were resistant against tetracycline. However, none of the LA-MRSA isolates tested was resistant against tigecycline. Conversely, we found tigecycline resistance in three of the HA-MRSA strains when incubated on Mueller-Hinton agar plates. Because of the recently reported influence of the test conditions on the in vitro susceptibility of tigecycline, we also tested a subset of the strains on Iso-Sensitest medium [START_REF] Fernández-Mazarrasa | High concentrations of manganese in Mueller-Hinton agar increase MICs of tigecycline determined by Etest[END_REF]. Eleven MRSA strains with the highest MICs for tigecycline were selected and retested on Mueller-Hinton agar and on Iso-Sensitest medium. On Mueller-Hinton agar, the results were identical to the initial

MIC on Mueller-Hinton MIC on Iso-Sensitest

Fig. 1 Linear regression analysis of MIC values for tigecycline using Mueller-Hinton medium and Iso-Sensitest medium result, but on Iso-Sensitest medium, the MICs for tigecycline were much lower, and all strains were considered to be susceptible. The results for tigecycline are influenced by the concentration of manganese in the medium [START_REF] Fernández-Mazarrasa | High concentrations of manganese in Mueller-Hinton agar increase MICs of tigecycline determined by Etest[END_REF]. As Mueller-Hinton agar is a biological medium, the concentration of manganese may vary. Iso-Sensitest is a biochemical medium, which is well-defined. However, the CLSI standard recommends the use of Mueller-Hinton medium for the susceptibility testing of tigecycline using the Etest system [START_REF]Methods for dilution antimicrobial susceptibility tests for bacteria that grow aerobically[END_REF]. This discrepancy requires further investigations into the underlying mechanisms.

An interesting aspect of this study is the remarkable difference in resistance against various classes of antibiotics between old and more recent strains of MRSA. The older strains were, in general, much more resistant than the more recent strains (Table 1). This may reflect the emergence of CA-MRSA in recent years, which are, in general, more susceptible [START_REF] Kluytmans-Vandenbergh | Communityacquired methicillin-resistant Staphylococcus aureus: current perspectives[END_REF]. The only antimicrobial agent with significantly higher MICs in recent MRSA was vancomycin. This has recently been reported by other groups and may reflect the increased use of this agent in hospitals all over the world [20,[START_REF] Wang | Increased vancomycin MICs for Staphylococcus aureus clinical isolates from a university hospital during a 5-year period[END_REF]. As vancomycin is considered to be the cornerstone of therapy for serious MRSA infections, the increasing MICs are a worrying finding. It stresses the need for alternative therapeutic agents. The LA-MRSA strains were also relatively susceptible to many classes of antibiotics, with the exception of tetracycline. MICs for tigecycline were comparable in all three groups of strains.

In conclusion, tigecycline exhibited broad in vitro activity against a collection of MRSA strains collected in The Netherlands, including livestock-associated strains. Using the recommended methodology, we found three strains to be resistant. However, these strains were considered to be susceptible when Iso-Sensitest medium was used. This discrepancy warrants further investigations into the preferred test conditions because the interpretation of the in vitro susceptibility of tigecycline is affected significantly.

Table 1

 1 MIC 50 and MIC 90 values of antimicrobial agents against 76 MRSA collected between 1990 and 1998 (old MRSA), 93 MRSA collected between 2003 and 2005 (recent MRSA), and 33 livestock-associated MRSA (LA-MRSA) strains collected in The Netherlands between 2003 and 2005

	Antibiotic	Old MRSA		Recent MRSA	LA-MRSA		p-value
		MIC 50	MIC 90	MIC 50	MIC 90	MIC 50	MIC 90	Old vs. recent	Old vs. LA	Recent vs. LA
	Tigecycline	0.19	0.42	0.19	0.38	0.25	0.38	0.618	0.230	0.296
	Oxacillin	256	256	32.0	256	12.0	48.0	<0.001 a	<0.001 a	<0.001 a
	Gentamicin	12	256	0.75	41.6	0.38	9.9	<0.001 a	<0.001 a	<0.001 a
	Cotrimoxazole	0.13	32	0.025	0.525	0.02	0.13	<0.001 a	<0.001 a	0.202
	Ciprofloxacin	32	32	24.0	32.0	0.38	1.8	<0.001 a	<0.001 a	<0.001 a
	Erythromycin	256	256	0.38	256	0.25	256	<0.001 a	<0.001 a	0.036
	Clindamycin	0.19	256	0.09	256	0.06	256	<0.001 a	<0.001 a	0.179
	Rifampin	0.012	32	0.006	0.60	0.004	0.006	<0.001 a	<0.001 a	<0.001 a
	Daptomycin	0.38	0.75	0.38	0.75	0.13	0.19	0.243	<0.001 a	<0.001 a
	Tetracycline	12	32	0.38	29.6	32	48	0.001 a	<0.001 a	<0.001 a
	Linezolid	1.0	1.5	1.0	1.5	0.75	1.0	0.032	<0.001 a	<0.001 a

Acknowledgments This study was financially supported by Wyeth Pharmaceuticals.

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