Zaki Leghtas

Gabriel Turinici

Herschel Rabitz

Pierre Rouchon

Hamiltonian identification through enhanced observability utilizing quantum control

This note considers Hamiltonian identification for a controllable quantum system with nondegenerate transitions and a known initial state. We assume to have at our disposal a single scalar control input and the population measure of only one state at an (arbitrarily large) final time T. We prove that the quantum dipole moment matrix is locally observable in the following sense: for any two close but distinct dipole moment matrices, we construct discriminating controls giving two different measurements. This result suggests that what may appear at first to be very restrictive measurements are actually rich for identification, when combined with well designed discriminating controls, to uniquely identify the complete dipole moment of such systems.

I. INTRODUCTION

Quantum control has been receiving increasing attention [START_REF] Brif | Control of quantum phenomena: past, present and future[END_REF] and one of its promising applications is to Hamiltonian identification [START_REF] Warren | Coherent control of quantum dynamics: the dream is alive[END_REF] by using the ability to actively control a quantum system as a means to gain information about the underlying Hamiltonian governing its dynamics. The underlying premise is that controls may be found which make the measurements not only robust to noise but also highly sensitive to the unknown parameters in the Hamiltonian. Hence, although the performance of laboratory measurements may be constrained, the ability to control a quantum system has the prospect of turning this data into a rich source of information on the system's Hamiltonian.

In this note, we consider the problem of identifying the dipole moment (which is assumed to be real) of an N -level quantum system, initialized to a known state (ground state), from a single population measurement at one arbitrarily large time T . We suppose an ability to freely control the system with a time dependent electric field (t). The measurements are obtained by (i) initializing at time t = 0 the system's state at a known state |i , (ii) controlling in open loop and without measurement the system with an electric field k (t) for t ∈ [0, T ] where T > 0, and (iii) measuring at final time T the population of one state |f . This may be repeated for many controls ( k ) k . We prove the existence of controls which make the identification from one population measurement a well posed problem (theorem 1). These controls have a simple physical interpretation in analogy with Ramsey interferometry (see Fig. 1).

The perspective above combined with control theory is motivated by three practical arguments. First, measuring a state population at one time T is a technique which can have a very high signal to noise ratio (∼ 100). Second, technological progress with spatial light modulators (SLM) permits generating a broad variety of controls in the laboratory. Third, ultra short pulsed fields can be well measured in the laboratory [START_REF] Iaconis | Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses[END_REF]. Hence, we are able to design a variety of precisely known control inputs.

Le Bris and al [START_REF] Lebris | Hamiltonian identification for quantum systems: well-posedness and numerical approaches[END_REF] prove the observability of the dipole moment when the population of all states are measured over an arbitrarily large interval of time. Algorithms to reconstruct the dipole from the measured data were proposed using nonlinear observers [START_REF] Leghtas | Parameter estimation of a 3-level quantum system with a single population measurement[END_REF], [START_REF] Bonnabel | Observer-based Hamiltonian identification for quantum systems[END_REF]. A different setting is considered in [START_REF] Schirmer | Quantum system identification: Hamiltonian estimation using spectral and Bayesian analysis[END_REF], [START_REF] Schirmer | Quantum system identification by Bayesian analysis of noisy data: Beyond Hamiltonian tomography[END_REF] where it is supposed that one can prepare and measure the system in a set of orthogonal states at various times, and the available data is the probability to measure the system in a certain state when it was prepared in another; Bayesian es-timation is used to reconstruct the energy levels, the damping constants and the dipole moment from the measured data. We consider here the less demanding case where the only available measurement is the population of one state at one arbitrarily large time, and the initial state is known and coincides with the ground state.

The note is organized as follows. In section II we state the main result in Theorem 1, and section III gives the proof of the Theorem and an important lemma on which the main result is based. Finally concluding remarks are presented in Section IV.

II. OBSERVABILITY OF THE QUANTUM DIPOLE MOMENT

A. Problem setting

We consider a quantum system in a pure state described by the wave function |ψ ∈ S. Here S is the set of N dimensional complex vectors of unit norm. The system interacts with an electric field (the real control input) ∈ E T for some T > 0 with E T {f : [0, T ] → R |f piecewise continuous}. For a given control we measure the population of the state |f at time T denoted as P if ( ). We denote by H 0 the free Hamiltonian (Hermitian matrix) and by µ the dipole moment operator, also a Hermitian matrix. The initial state |i and the measured state |f are eigenvectors of H 0 . We consider a semiclassical model for the light-matter interaction, and the dynamics of |ψ are given by the Schrodinger equation:

ı ∂ ∂t |ψ(t) = (H 0 -(t)µ) |ψ(t) (1)
|ψ(0) = |i , P if ( ) = | f |ψ(T ) | 2 .
For all T > 0, we suppose that we can create any field ∈ E T and that we can measure P if ( ). For M different fields { 1 , .., M } we can collect the measurements {P if ( 1 ), .., P if ( M )}. Through (1) P if is a function of µ and a functional of , and when necessary this explicit dependence will be written as P if ( , µ). The aim of this note is to explore the feasibility of estimating the dipole moment µ from the measured data {P if ( 1 ), .., P if ( M )} using well chosen controls { 1 , .., M }. Below, P if ( , µ) refers to the measurement achieved on the real system using a control , and for any μ, P if ( , μ) is the estimated measurement which is obtained by simulating system (1) with the control and coupling μ.

B. Main result

For all k ≤ N we denote |k as the eigenvector of H 0 with associated eigenvalue E k . Throughout the note, all matrices are written in the basis (|1 , .., |N ). The initial and measured states correspond to some indexes i, f ∈ {1, . . . , N }. For all k, l ≤ N we specify σ x lk |l k|+|k l|. We define

M Span{σ lk x \k, l ≤ N with Tr µσ lk x = 0},
with Tr being the trace operation. When all non diagonal elements of µ are non-null,

M = dim(M) = N (N -1) 2
. The main result is the following:

Theorem 1. Consider a real symmetric matrix µ with zero diagonal entries and a real diagonal matrix H 0 with non-degenerate transitions. Suppose that the system state in (1) is controllable. Then for any positive constant α there exists a time T > 0 and M fields ( 1 , .., M ) ∈ E T M such that the cost function

J : M μ → M k=1 (P if ( k , μ) -P if ( k , µ)) 2 is in C 2 (M, R) and locally α-convex 1 around µ.
C k (A, B) denotes the set of k times continuously differentiable functions defined over A with values in B. In the appendix we provide the definitions of controllability and a matrix with non-degenerate transitions. Here and throughout this note, the norm of matrix µ, noted µ refers to the max norm. A direct consequence of Theorem 1 is the local observability of the dipole moment: Corollary 1. Under the assumptions of Theorem 1, the dipole moment is locally observable in M. a) Proof: Take α > 0. Theorem 1 implies that there exists a time T > 0 and M fields ( 1 , .., M ) ∈ E T M such that the cost function J is C2 (M, R) and locally α-convex around µ. Hence there ∃r > 0 such that for all μ ∈ M with μ -µ ≤ r and μ = µ, J(μ) > 0, and hence there exists

∈ { 1 , .., M } such that P if ( , μ) -P if ( , µ) = 0.
Remark 1. The local α-convexity is a property stronger than the mere possibility to identify the dipole matrix. It states that the distinction between a dipole candidate μ and the true dipole µ can be observed (through the measurements aggregated in J) to first order in the distance µ -μ . This first order dependence of the measurement P if with respect to the dipole µ is addressed in more detail in lemma 1. For well chosen controls, the J function has a very simple shape around µ and a simple gradient algorithm could be used to identify it.

The eigenvalues of H 0 are commonly measured through spectroscopy and can be found in reference tables [START_REF]) data sheets[END_REF] with precisions of order 10 -7 . The result of theorem 1 is also relevant for the problem of discriminating between two molecules with the same free Hamiltonian and different effective dipole operators. In that framework, μ and µ would be the dipole operators of these two molecules (as opposed to one estimated and one true dipole, as considered in this note), and the aim is to find controls which produce different data sets for these two different but similar quantum systems. This was experimentally accomplished in [START_REF] Petersen | How shaped light discriminates nearly identical biochromophores[END_REF] where a genetic algorithm is used to find these discriminating controls. A complementary theoretical controllability analysis can be found in [START_REF] Turinici | Optimal discrimination of multiple quantum systems: controllability analysis[END_REF].

III. PROOFS

A. Existence of discriminating controls

We denote µ = 1 µ µ the normalized dimensionless dipole moment operator, µ lk Tr µ σ lk x and ∂P if ∂µ lk ( ) the partial derivative of P if ( ) with respect to µ lk . Theorem 1 is based on the following lemma: Lemma 1. Suppose that µ is real, symmetric and has only zeros on its diagonal and H 0 is real, diagonal, with non-degenerate transitions. Suppose system (1) is controllable. Then for all (l, k) with µ lk = 0, there exists ξ 0 > 0 such that, for all ξ ∈]0, ξ 0 [, exist T > 0 and ∈ E T satisfying

• ∂P if ∂µ lk ( ) = 1 2ξ + O(1) • ∀{m, n} = {l, k} with µ mn = 0, ∂P if ∂µ mn ( ) = O(1),
where O(1) corresponds to zero order terms with respect to ξ around 0 + .

B. Proof of Theorem 1

To each pair of integers (l p , k p ), l p < k p such that Tr µ σ

> 0 such that ∀ξ ∈ ]0, ξ 0 [, ∃ T 1 , .., T M and ( 1 , .., M ) ∈ E T 1 × .. × E T M such that: (i) ∀p ∈ [1 : M ] ∂P if ∂µ p ( p ) = 1 2ξ + O(1) and (ii) ∀p = p ∂P if ∂µ p ( p ) = O(1)
. We take T = max(T 1 , .., T M ) and for all k ∈ {1, .., M } we extend the definition of k from [0, T k ] to [0, T ] by taking k (t) = 0 for all t ∈]T k , T ]. We will use J : M → R defined by:

J(μ) = M k=1 (P if ( k , μ) -P if ( k , µ)) 2 .
For a fixed T > 0 and

∈ E T , since µ → P if ( , µ) is C 1 (M, R), J(μ) is in C 2 (M, R). We find ∂ 2 J ∂µ p ∂µ p (µ) = M k=1 ∂P if ∂µ p ( k , µ) ∂P if ∂µ p ( k , µ) so that for all p ∈ {1, ..., M } : ∂ 2 J ∂µ p 2 (µ) = 1 4ξ 2 + O( 1 ξ ) and when p = p ∂ 2 J ∂µ p ∂µ p (µ) = O( 1 ξ ). We have: ∇ 2 J(µ) = 1 4ξ 2 (I + O(ξ))
, where ∇ 2 J(µ) is the Hessian of J at µ and I is the identity matrix. The smallest eigenvalue of ∇ 2 J(µ) scales as 1 4ξ 2 (1 + O(ξ)), hence by taking ξ small enough it can be made larger than α thereby reaching the conclusion above.

C. Proof of lemma 1

We define the dimensionless time scale τ 1 H 0 t and also 1 H 0 T . For two times τ, τ ∈ [0, ] we define the propagator U (τ , τ ) such that |ψ(τ ) = U (τ , τ ) |ψ(τ ) . Rewriting [START_REF] Brif | Control of quantum phenomena: past, present and future[END_REF] for U (τ, 0) we obtain:

ı ∂ ∂τ U (τ, 0) = 1 H 0 (H 0 -(τ )µ)U (τ, 0) (2)
P if ( ) = | f | U ( , 0) |i | 2 , U (0, 0) = I .
The proof of lemma 1 has two parts I and II separately treated below.

i) Part I: Take two times τ 1 , τ 2 , 0 < τ 1 < τ 2 < . We can write (for any complex z we denote by z its complex conjugate):

P if ( ) = z z where z = f | U ( , τ 2 )U (τ 2 , τ 1 )U (τ 1 , 0) |i .
Denote for any m, n = 1, ..., M : ω mn Em-En H 0 and consider the control defined on [τ 1 , τ 2 ]:

(τ ) = ε cos(ω lk (τ -τ 1 )) , (3)
where ε is a small strictly positive real parameter. Take ξ = µ ε H 0 . The only remaining degree of freedom in the control over [τ 1 , τ 2 ] is ξ, which can be taken arbitrarily small. We define H 0 = 1 H 0 H 0 and ω mn = m| H 0 |m -n| H 0 |n . Note that ω mn = -ω nm . We have [START_REF] Beltrani | Photonic reagent control of dynamically homologous quantum systems[END_REF]

∂ ∂µ lk U (τ 2 , τ 1 ) = ı µ H 0 U (τ 2 , τ 1 ) × τ 2 τ 1 (τ )U † (τ, τ 1 )σ lk x U (τ, τ 1 )dτ . ( 4
)
We now rewrite (2) and ( 4) for the control given in (3) on the time interval [τ 1 , τ 2 ]:

ı ∂ ∂τ U (τ, τ 1 ) = (H 0 -ξ cos(ω lk (τ -τ 1 ))µ )U (τ, τ 1 ) (5) ∂ ∂µ lk U (τ 2 , τ 1 ) = ıξU (τ 2 , τ 1 ) × τ2 τ1 cos(ω lk (τ -τ 1 ))U † (τ, τ 1 )σ lk x U (τ, τ 1 )dτ . ( 6
)
The goal is to show that ∂ ∂µ lk U (τ 2 , τ 1 ) can be made arbitrarily "large" while ∂ ∂µ mn U (τ 2 , τ 1 ) stays bounded. Note that all the terms in the integrand of ( 6) are bounded, and a rough estimate of the norm of ∂ ∂µ lk U (τ 2 , τ 1 ) gives a quantity proportional to (τ 2 -τ 1 )ξ. Hence, we take τ 2 -τ 1 = 1 ξ 2 , implying the need to have expressions for U (τ, τ 1 ) over a time scale on the order of 1 ξ 2 . To this end we state lemma 2 which gives such an approximation. Lemma 2. Consider Eq. ( 5). There exists a Hermitian matrix K and ξ 0 > 0 such that, for any ξ ∈]0, ξ 0 [, we have:

sup τ ∈[τ1,τ1+ 1 ξ 2 ]
U (τ, τ 1 ) -e -ıH 0 (τ -τ1) e ı(ξ µ lk

2 σ lk x +ξ 2 K)(τ -τ1) = O(ξ).
We continue with the proof of Lemma 1 and will come back to Lemma 2 in Section III-D.

Using the expression of U (τ, τ 1 ) given in lemma 2, the integrand in ( 6) is:

cos(ω lk (τ -τ 1 ))U † (τ, τ 1 )σ lk x U (τ, τ 1 ) = e -ı(ξ µ lk 2 σ lk x +ξ 2 K)(τ -τ1) (cos(ω lk (τ -τ 1
))e ıH 0 (τ -τ1)

σ lk x e -ıH 0 (τ -τ1) )e ı(ξ µ lk

2 σ lk x +ξ 2 K)(τ -τ1) + O(ξ) .
In order to compute ( 6), we need the following result:

cos(ω lk (τ -τ 1 ))e ıH 0 (τ -τ1) σ lk x e -ıH 0 (τ -τ1) = 1 2 σ lk x + 1 2 cos(2ω lk (τ -τ 1 ))σ lk x + 1 2 sin(2ω lk (τ -τ 1 ))σ lk y , (7)
where we denote σ lk y = +ı |l k| -ı |k l|. In ( 7), the terms oscillating at frequency 2ω lk independent of ξ will only contribute to O(ξ) in [START_REF] Bonnabel | Observer-based Hamiltonian identification for quantum systems[END_REF]. We now focus on the contribution of the term with σ lk x in (6) which calls for (see appendix): ∀τ e -ı(ξ µ lk

2 σ lk x +ξ 2 K)(τ -τ 1 ) σ lk x e ı(ξ µ lk 2 σ lk x +ξ 2 K)(τ -τ 1 ) = σ lk x + O(ξ) . (8)
Introducing ( 8) into (6), we find:

∂ ∂µ lk U (τ 2 , τ 1 ) = ıξU (τ 2 , τ 1 ) τ 2 -τ 1 2 σ lk x + O(1) + (τ 2 -τ 1 )O(ξ) .
From now on, we take τ 2 = τ 1 + 1 ξ 2 and obtain:

∂ ∂µ lk U (τ 2 , τ 1 ) = ıU (τ 2 , τ 1 )( 1 2ξ σ lk x + O(1)). (9)
We define |ψ 1 |l and |ψ 2 1 √ 2 U (τ 2 , τ 1 )(|l + ı |k ). Since the system is controllable there exists a time τ 1 and a field 1 ∈ E τ 1 such that U (τ 1 , 0) |i = |ψ 1 , and there exists a time and a field 2 defined over [τ 2 , ] such that U † ( , τ 2 ) |f = |ψ 2 . Since the state space is compact (here it is a sphere), we know that if the system is controllable, it is controllable in bounded time, and with bounded controls (see Theorem 6.5 in [START_REF] Jurdjevic | Control systems on Lie groups[END_REF]). Hence, -τ 2 can be chosen bounded for all ξ. Therefore ∂ ∂µ lk U (0, τ 1 ) and ∂ ∂µ lk U (τ 2 , ) are bounded. Thus, we have:

∂ ∂µ lk P if ( ) = 2 ( f | U ( , τ 2 ) ∂ ∂µ lk U (τ 2 , τ 1 )U (τ 1 , 0) |i i| U † (τ 1 , 0)U † (τ 2 , τ 1 )U † ( , τ 2 ) |f ) + O(1).
We now utilize U (τ 1 , 0) |i = |ψ 1 and U † ( , τ 2 ) |f = |ψ 2 where |ψ 1 and |ψ 2 are defined above, and replace ∂ ∂µ lk U (τ 2 , τ 1 ) by its expression in [START_REF]) data sheets[END_REF] to find: ∂ ∂µ lk

P if ( ) = 1 2ξ + O(1)
. This expression holds for the control defined as:

(τ ) =    1 (τ ), if τ ∈ [0, τ 1 ] H 0 µ ξ cos(ω lk (τ -τ 1 )), if τ ∈]τ 1 , τ 2 [ 2 (τ ), if τ ∈ [τ 2 , ] (10)
ii) Part II: We now need to prove that

∂ ∂µ mn P if ( ) = O(1)
for {m, n} = {l, k}, where is the control found above in [START_REF] Petersen | How shaped light discriminates nearly identical biochromophores[END_REF]. As in Eq. ( 6), we have:

∂ ∂µ mn U (τ 2 , τ 1 ) = ıξU (τ 2 , τ 1 )× τ2 τ1 cos(ω lk (τ -τ 1 ))U † (τ, τ 1 )σ mn x U (τ, τ 1 )dτ , (11)
and again the result of lemma 2 is employed. Eq. ( 11) calls for 2 cos(ω lk (τ -τ 1 ))e ıH 0 (τ -τ1) σ mn x e -ıH 0 (τ -τ1) = cos((ω lk -ω mn )(τ -τ 1 ))σ mn x -sin((ω lk -ω mn )(τ -τ 1 ))σ mn y + cos((ω lk +ω mn )(τ -τ 1 ))σ mn x +sin((ω lk +ω mn )(τ -τ 1 ))σ mn y .

Considering that H 0 has non-degenerate transitions (see definition in the appendix) implies that ω lkω mn = 0 and ω lk + ω mn = 0. As the expression in [START_REF] Beltrani | Photonic reagent control of dynamically homologous quantum systems[END_REF] oscillates at frequencies independent of ξ, it therefore contributes to O(ξ) in [START_REF] Turinici | Optimal discrimination of multiple quantum systems: controllability analysis[END_REF]. Hence, for τ 2τ 1 = 1 ξ 2 we can directly conclude that ∂ ∂µ mn

P if ( ) = O(1).

D. Proof of lemma 2

This proof relies on three consecutive changes of frame that aim to cancel the oscillating terms of order 0 and 1 with respect to ξ. We then derive a specific form of the averaging Theorem (see theorem 4.3.6 in [START_REF] Sanders | Averaging methods in nonlinear dynamical systems[END_REF] for a general form of the averaging theorem). For the sake of clarity and with no loss of generality, we take τ 1 = 0 and note

U (τ )
U (τ, τ 1 ). Eq. ( 5) may be written in the interaction frame U I (τ ) e ıH 0 τ U (τ ),

∂ ∂τ U I (τ ) = ıξ µ lk 2 σ lk x + ∂ ∂τ H I (τ ) U I (τ )
where:

∂ ∂τ H I (τ ) = 1 2 (m,n) =(k,l)
µ mn e ı(-ω kl +ω mn )τ |m n|

+ 1 2 (m,n) =(l,k)
µ mn e ı(-ω lk +ω mn )τ |m n| , and the average of H I is zero. The average of a time dependent operator C(τ ) is defined as follows (see defintion 4.2.4 in [START_REF] Sanders | Averaging methods in nonlinear dynamical systems[END_REF]): C = lim θ→+∞ 1 θ θ 0 C(τ )dτ . We now take U I (τ ) = (I -ıξH I (τ ))U I (τ ). Since ∂ ∂τ H I is almost periodic 2 , then H I is also almost periodic and hence bounded for all τ . Hence, there exists ξ 0 > 0, ∀ξ < ξ 0 , I -ıξH I (τ ) has an inverse and (I -ıξH I (τ )) -1 = I + ıξH I (τ ) + O(ξ 2 ). We find:

∂ ∂τ U I (τ ) = ı ξ µ lk 2 σ lk x - ıξ 2 µ lk 2 [H I (τ ), σ lk x ] + H I (τ ) ∂ ∂τ H I (τ ) + O(ξ 3 ) U I (τ ).
Notice that, with K = -ıH I ∂ ∂τ H I independent of ξ and K(τ ) almost periodic with zero average, we also have:

µ lk 2 [H I (τ ), σ lk x ]+H I (τ ) ∂ ∂τ H I (τ ) = ı(K + ∂ ∂τ K(τ )). It is important to note that 1 2 ∂ ∂τ H 2 I = 0 = H I ∂ ∂τ H I + ( ∂ ∂τ H I )H I = ı(K -K † ). Hence K = K † is Hermitian.
We now take U I (τ ) = (I -ıξ 2 K(τ ))U I (τ ). Since K(τ ) is bounded for all τ , then for a sufficiently small ξ, Iıξ 2 K(τ ) has an inverse and (I -ıξ 2 K(τ )) -1 = I +ıξ 2 K(τ )+O(ξ 4 ). U I satisfies the following equation:

∂ ∂τ U I (τ ) = ı ξ µ lk 2 σ lk x + ξ 2 K + O(ξ 3 ) U I (τ ) , (13)
and we define U av to be the solution to the averaged dynamics (U av (0) = I):

∂ ∂τ U av (τ ) = ı ξ µ lk 2 σ lk x + ξ 2 K U av (τ ) . ( 14
)
We can directly solve [START_REF] Sanders | Averaging methods in nonlinear dynamical systems[END_REF]:

U av (τ ) = e ı ξ µ lk 2 σ lk x +ξ 2 K τ
. Subtracting ( 13) from ( 14), we find, using Gronwall's lemma, that for all τ < 1 ξ 2 one has U I (τ ) = U av (τ ) + O(ξ). Also note that to go from U I to U I we have used two 2 Can be written as M k=1 e iω k τ A k consecutive changes of variables which are close to the identity, hence: ∀τ U I (τ ) = U I (τ ) + O(ξ).

Finally, since e -ıH 0 τ is an isometry, we have:

U (τ ) = e -ıH 0 τ e ı(ξ µ lk 2 σ lk x +ξ 2 K)τ + O(ξ) for all τ ≤ 1 ξ 2 .
IV. CONCLUSION Identification of the real dipole moment matrix is shown to be well posed for a controllable finite dimensional quantum system with non-degenerate transitions and using as measurements only one population at a final time T . The results also provide a theoretical foundation to optimal discrimination experiments. APPENDIX Definition 1. We say that system (1) is controllable [START_REF]Introduction to Quantum Control and Dynamics[END_REF] if for all |ψ 1 , |ψ 2 ∈ S there exists a time t and a control ∈ E t such that for |ψ(0) = |ψ 1 , (1) leads to |ψ(t) = |ψ 2 . Definition 2. Let H 0 and µ be N × N Hermitian matrices. We denote E 1 , .., E N the eigenvalues of H 0 and |1 , .., |N its corresponding eigenvectors. We say that H 0 has non-degenerate transitions [START_REF] Turinici | Wavefunction controllability for finite-dimensional bilinear quantum systems[END_REF] if ∀(l, k) = (m, n), l = k and m = n, such that l| µ |k = 0 and m| µ |n = 0, we have E l -E k = E m -E n .

Definition 3. Take system (1). Let us denote M as the space to which µ belongs. We say that µ is locally observable in M if there exists r > 0 such that for all μ ∈ M with 0 < μ-µ ≤ r there exists T > 0 and ∈ E T such that P if ( , μ) = P if ( , µ).

Computation: Here, we compute Σ lk x (τ ) = e -i(ξ µ lk 2 σ lk x +ξ 2 K)(τ -τ 1 ) σ lk x e i(ξ µ lk 2 σ lk x +ξ 2 K)(τ -τ 1 ) . We have µ lk = 0 and σ lk x + ξ 2K µ lk is Hermitian. Hence, there exists a unitary matrix P ξ and a real diagonal matrix ∆ ξ such that σ lk x + ξ 2K

  associate a unique index p ∈ {1, ..., M }, and we define σ p x σ lpkp x along with µ p Tr (µ σ p x ). According to lemma 1, ∃ξ 0

Fig. 1 .

 1 Fig. 1. A good control has three components (inspired from Ramsey interferometry) to enable the identification of µ lk . The field 1 is defined over [0, τ1] (analog of first Ramsey pulse) to steer the known initial state |i to |ψ1 = |l : |l = U (τ1, 0) |i . The field 2 is defined over [τ2, ] (analog of second Ramsey pulse) is such that |f = U ( , τ2) |ψ2 where |ψ2 = U (τ2, τ1) |l +ı|k √ 2 and the propagator U (τ2, τ1) corresponds, for a long interval τ2 -τ1, to a large number of Rabi oscillations with the control ε cos(ω lk (τ -τ1)) resonant with the |l ↔ |k transition.

=

  P ξ ∆ ξ P † ξ . The function ξ ∈ [0, ξ 0 ] → σ lk x + ξ 2K µ lkis analytic, therefore the eigenvectors of σ lk x + ξ 2K µ lk can be continued analytically as a function of ξ (see Theorem 6.1 in chapter II, §6 section 1 and 2 in[START_REF] Kato | Perturbation theory for linear operators[END_REF]). Hence, P ξ = P 0 + O(ξ) where P 0 is such that P † 0 σ lk x P 0 = σ lk z is real and diagonal. σ lk z = |l l| -|k k|. We find, ∀τ :Σ lk x (τ ) = σ lk x + O(ξ), where O(ξ) is a first order term in ξ and a bounded function of τ .

The smallest eigenvalue of the Hessian ∇

J(µ) is larger than α.

ACKNOWLEDGMENT

We thank Mazyar Mirrahimi and Alexei Goun for discussions. ZL acknowledges support from Agence Nationale de la Recherche (ANR), Projet Jeunes Chercheurs EPOQ2 number ANR-09-JCJC-0070. GT was supported by the CNRS through a PICS grant. The authors were partially supported by the ANR, Projet Blanc EMAQS number ANR-2011-BS01-017-01, and by ANR, Projet C-QUID BLAN-3-139579. We acknowledge support from the US department of energy.


Bettina Migge

Susanne Mühleisen

Earlier Caribbean English and Creole in writing

In research on Creoles, historical written texts have in recent decades been fruitfully employed to shed light on the diachronic development of these languages and the nature of Creole genesis. They have so far been much less frequently used to derive social information about these communities and to improve our understanding of the sociolinguistics and stylistic structure of these languages. This paper surveys linguistic research on early written texts in the anglophone Caribbean and takes a critical look at the theories and methods employed to study these texts. It emphases the sociolinguistic value of the texts and provides some exemplary analyses of early Creole documents.

Introduction

Research on Caribbean English-lexicon Creoles has made use of early textual documents collected by missionaries (cf. for instance, [START_REF] Oldendorp | [END_REF]Oldendorp [1767Oldendorp -1768]], 2000 [1777]; cf. also [START_REF] Mühlhäusler | Die Rolle der Missionare in Hugo Schuchardts kreolischen Studien[END_REF], planters (e.g. [START_REF] Leslie | A New History of Jamaica[END_REF][START_REF] Lewis | Journal of a West India Proprietor, kept during a Residence in the Island of Jamaica[END_REF][START_REF] Long | History of Jamaica, or General Survey of the Ancient and Modern State of that Island: With Reflections on its Situations, Settlements, Inhabitants, Climate, Products Comerce, Laws and Government[END_REF][START_REF] Long | History of Jamaica, or General Survey of the Ancient and Modern State of that Island: With Reflections on its Situations, Settlements, Inhabitants, Climate, Products Comerce, Laws and Government[END_REF]), travellers (e.g. [START_REF] Nugent | Lady Nugent's Journal: Jamaica One Hundred Years Ago[END_REF][START_REF] Nugent | Lady Nugent's Journal: Jamaica One Hundred Years Ago[END_REF]; Sloane 1707) or fictional accounts by writers (e.g. [START_REF] Scott | Tom Cringle's Log[END_REF]. Historical written texts continue to be an important source for shedding light on earlier stages of creolization (e.g. Arends 1995aArends , 1995b;;[START_REF] Arends | Early Creole Texts : A Collection of Eighteenth-Century Sranan and Saramaccan Documents[END_REF]Baker and Bruyn 1999), the diachronic development of Creoles as well as on structural features and variation within a particular variety. The size of text corpora varies greatly within the Caribbean: while there is a very substantial collection of earlier texts on Sranan Tongo, Saamaka and Negerhollands (cf. Arends 2002: 50). Textual documentation of other early Caribbean Englishes and Creoles is less prolific and consists of diverse text types -from short quotations of speech contained within travel accounts of outsiders to whole texts written in Creole by insiders. For instance, lyrics of songs, poems, cartoon captions or commentaries that appeared in newspapers for the aim of entertainment (cf. Baker and Winer 1999: 103-4; for an overview of the development of literate culture in the Caribbean, cf. [START_REF] Roberts | From Oral to Literate Culture. Colonial Experience in the English West Indies[END_REF].

Collections of eighteenth-and nineteenth-century texts on Jamaican (D'Costa and Lalla 1989;[START_REF] Lalla | Language in Exile: Three Hundred Years of Jamaican Creole[END_REF], Trinidadian [START_REF] Winer | Early Trinidadian English Creole: The Spectator Texts[END_REF][START_REF] Winer | Suzie and Sambo: Linguistic archaism or working class solidarity[END_REF][START_REF] Winer | Penny Cuts: Differentiation of varieties in Trinidad English Creole, 1904-06[END_REF][START_REF] Winer | Six vernacular texts from Trinidad, 1839-1851[END_REF][START_REF] Winer | Language varieties in early Trinidadian novels, 1838-1907[END_REF], Guyanese (Rickford 1987: 81-120) or Bajan [START_REF] Rickford | Textual evidence on the nature of early Barbadian speech 1676-1835[END_REF][START_REF] Fields | Early Bajan: Creole or Non-Creole?[END_REF]) have also been less subject to structural research than their Surinamese counterparts. An exception to this is research on early Kittian texts to which a whole volume (Baker and Bruyn 1999) is devoted. Recent years have seen a wealth of new and annotated editions of nineteenth century Caribbean fiction, 1 where embedded speech portrayal may also serve as useful language material. The time period of text collections covered comprises mainly the eighteenth and the nineteenth century, even though isolated material from the seventeenth century can also be found for early Bajan (Rickford and Handler 1994: 223-229;[START_REF] Fields | Early Bajan: Creole or Non-Creole?[END_REF].

It is clear that writing is never an accurate representation of speech, neither in its standardised form nor -unquestionably -in the highly flexible recordings of colonial writings that were greatly influenced by the documenter's own linguistic background, linguistic knowledge and attitude towards the subject of recording. Due to the social conditions in the Caribbean plantation society, early texts in the Caribbean are less likely to have been produced by actual speakers of the language varieties (Creole/Caribbean Englishes) than in any other region covered by this volume. Several authors (e.g. Rickford 1987: 81;Baker and Winer 1999: 114) have remarked on the high proportion of non-native authors of such texts. To use earlier Caribbean English and Creole written texts as a source for linguistic reconstructions and feature analysis, the researcher must therefore exercise necessary caution and consider not only the historical background of the language situation in which it was produced (cf. Arends 1995b: 54), but also additional features such as linguistic and personal background of the writer, text type, writing event and intended audience. In addition to the linguistic information that can be filtered from the documents, a wealth of text-and sociolinguistic information can also be obtained from them, e.g. literacy practices, the emergence of different text types and orthographies, but also attitudes towards language varieties and the social conditions of writing and text production. This contribution will not only give a survey of linguistic research on early written texts in the anglophone Caribbean but also take a critical look at the theories and methods employed to make use of them. It will thirdly also place emphasis on the sociolinguistic value of the texts and provide some exemplary analyses of early Creole documents. [START_REF] Warner-Lewis | Archibald Monteath: Igbo, Jamaican, Moravian[END_REF]. Archibald Monteath: Igbo, Jamaican, Moravian (reconstruction of a Caribbean slave narrative, with original material). [START_REF] Hall | Miserable Slavery. Thomas Thistlewood in Jamaica[END_REF]. In Miserable Slavery. Thomas Thistlewood in Jamaica, 1750-86 (reconstruction of a diary of a small penkeeper in mid eighteenth -century Jamaica).

The context of writing in the eighteenth-and nineteenth-century Caribbean

With the exception of two texts in Bajan (cf. Rickford and Handler 1994: 223-229) from the seventeenth century, the first materials written in Caribbean Englishlexicon Creoles date from the early part of the eighteenth century. The first materials for Sranan Tongo (henceforth Sranan), for instance, date from 1707 and the earliest records and reports for Jamaican Creole appear from the 1740s onwards. However, for all languages, the documents from the second part of the century are much more substantial, numerous and also socially diverse. In countries like Trinidad or Grenada records of English-lexicon Creoles emerged only in the early nineteenth century due to various colonial power and language shifts in the eighteenth century. In this section we discuss the characteristics of the available textual sources.

The authors

Social information on the authors of written materials in a Creole is highly variable. Sometimes the writers were quite well known, while in other cases there is no information and the social background has to be inferred from various sources. In the case of Suriname, there is relatively detailed information about the writer of the early Sranan and Saamaka dictionaries, C. L. Schumann (cf. Van den Berg 2007: 28). By contrast, the identity of the author of the Herlein fragment ( 1718) "remains somewhat of a mystery" (Van den Berg 2007: 19) and in the case of the Van Dyk language guide (c 1765), there is "not enough evidence […] to positively identify him" (Van den Berg 2007: 22). However, a good bit of social information about him can be gleaned from his writing (Arends 1995b;[START_REF] Bruyn | Grammaticalization in Creoles: The Development of Determiners and Relative Clauses in Sranan[END_REF]. The life and times of the author of the collection of early Kittian texts (1793, 1822), Sam Mathews, is subject to a whole chapter in the Baker and Bruyn volume (O'Flaherty 1999). Many of the West Indian planters whose diaries contain excerpts of Creole records stayed only for a short time and wrote from the point of view of an outsider. An exception is Matthew 'Monk' Lewis [START_REF] Lewis | Journal of a West India Proprietor, kept during a Residence in the Island of Jamaica[END_REF] due to his other career as a writer of fiction. The writers of Creole texts came from a relatively diverse set of backgrounds. In the case of Suriname and the Virgin Islands, for instance, writers had different national origins. Some of the 'Surinamese' authors came from the Netherlands (e.g. Herlein, Nepveu, van Dyk, Weygandt), while others such as Schumann, Wietz and Riemer were German. Suriname also boasts a number of local authors such as King, Focke, Kraag, Albitrouw, Alabi, Grego who were born and raised in Suriname. This suggests that writers came from different linguistic backgrounds, including writing conventions, and also had different levels of competence in the Creole. Only some of the authors, generally those who grew up in Suriname, were native speakers of one of the Creoles and used it as their main means of communication. However, this does not necessarily mean that they wrote in their native language but may have used a closely related language. For example, in the case of Johannes King, who wrote religious texts in Sranan, we know that he was exposed to the Maroon Creoles Matawai and Ndyuka by his parents, however, Sranan "belonged to his repertoire from early on, and even though it may not have been his mother tongue, it must have been a primary language for him" (Bruyn 1995: 37) because he partly grew up in Paramaribo. In colonial British West Indies, most of the authors of planters' diaries and travelogues came from the British motherland. Their linguistic ability to record the speech they heard varied considerably. J. B. Moreton, a bookkeeper in the Jamaican parish of Clarendon and collector of songs and stories (1793), was a non-native of Jamaica but "his record of songs and conversations suggest a keener and more sympathetic ear than that of any earlier commentator" (D'Costa and Lalla 1989: 12). In the nineteenth century, local writing gained ground in the Caribbean, not only because of the arrival of printing presses but also due to advances in education. [START_REF] Winer | Adolphus, A Tale (Anonymous) and The Slave Son by Mrs[END_REF] estimates that Adolphus, a novel by an anonymous author serialised in the Trinidadian newspaper in 1853 is "the first Trinidadian -and possibly Caribbean -novel by a presumably non-white writer born and raised in Trinidad" (2003: x).

The majority of the writers have to be classified as second language speakers with differences in linguistic competence. They usually learned the Creole later in life, often for professional reasons, and probably did not use it as their main means of communication. In the case of Suriname, some writers spent only relatively little time in the country and had comparatively little exposure to, and competence in, the different varieties at the time of writing (e.g. Riemer and Herlein). However, others such as Nepveu spent most of their life in the colony and are likely to have been fluent users in the language (Arends 1995b: 18). But time spent in the country is not always the best predictor for language competence. Although Schumann only stayed for eight years in Suriname, his writings are generally assumed to be very accurate, most likely because he had had linguistic training and was actively engaged in language description and translation projects using native consultants.

Authors were also professionally quite diverse. A good number of the eighteenth-and nineteenth-century documents were authored by Moravian Missionaries (e.g. Schumann and Wietz in Schuchardt 1914; Riemer 1780 in [START_REF] Arends | Early Creole Texts : A Collection of Eighteenth-Century Sranan and Saramaccan Documents[END_REF]cf. Brathwaite 2005: 256-7), but Wesleyan Methodists were also active in the Caribbean (cf. Brathwaite 2005: 345 on letters and Anon. "A series of tracts for Slaves in the West Indies"). Their educational backgrounds and abilities as linguists and translators must have varied considerably though. For instance, while Schumann appears to have studied theology in Germany and is generally described as a very skilled observer of language, Riemer who had previously worked as a weaver, lacked this scholarly background (Perl 1995: 247). Other writers held more mundane professions. For instance, Nepveu was a public servant who first worked as a secretary for the governor of Suriname and then became governor himself (Arends 1995b: 17). The 'authors' of the court records were scribes. Weygandt was an auctioneer of books and Van Dyk was possibly a slave officer on one of the plantations (Arends 1995b). Little is known about the professional background of the native speaker authors who mostly wrote during the nineteenth century. [START_REF] Bruyn | Grammaticalization in Creoles: The Development of Determiners and Relative Clauses in Sranan[END_REF] suggests that Focke was a lawyer and that Johannes King engaged in religious activities teaching the Gospel among the Ndyuka and in the Para region. Some of the other writers were engaged in prosylitizing but most likely also performed other activities such as farming. This suggests that the different authors had different degrees of familiarity, experience and ability in the medium of writing prior to their writing in a Creole. Some must have regularly engaged in writing in their own language(s) and had a good knowledge of writing conventions while others most likely had few opportunitieslacking in-depth knowledge of these conventions. In the West Indies, native speaker authors were faced with the fact that educational opportunity usually came with writing in the Creole's lexifier language, thus making it a difficult act to balance the conventions of English orthography with the marking of Creole phonological and grammatical features (see also 2.3 below).

The nature of the texts

The historical documents differ in two important respects: size and text type. Some sources are relatively small such as the Herlein fragment (1716) which consists of just 200 tokens while others such as Schumann's Sranan Dictionary or Van Dyk's (1765) language manual are substantial, numbering 20,000 and 15,000 tokens respectively (Van den Berg 2007: 18). 2The available texts also belong to different text categories. Using criteria such as a) the reality of a speech event portrayed, b) the relationship between the speaker and the recorder and c) the temporal distance between speech event and its recording, [START_REF] Schneider | Investigating variation and change in written documents[END_REF] typology of historical texts posits five categories of texts -recorded, recalled, imagined, observed, invented -that "represent a continuum of increasing distance between an original speech event and its written record" (2002: 72). The Caribbean documents include texts from all of these categories. For instance, Van den Berg (2007: 19) argues that the Saamaka Peace treaty (originally performed) and written in Sranan and the Sranan tokens from the judicial court records belong to what Schneider (2002: 73) classifies as "the most reliable and potentially the most interesting" type of data, so-called recorded data since they present a direct record of a speech event which actually occurred. For the West Indies, such direct recordings of texts are mainly found in artistic forms such as poems, lyrics of songs and tales (cf. D 'Costa and Lalla 1989;[START_REF] Lalla | Language in Exile: Three Hundred Years of Jamaican Creole[END_REF]Roberts 1997: 34-68). As a primarily oral format, such texts often follow rather formulaic and fixed patterns, mnemonic devices such as rhyme and collocations, making oral transmission over generations possible before they were written down. This also includes the possibility that such texts actually preserve earlier speech forms and archaisms. In addition to these artistic forms, there are also recorded texts from actual speech events, such as sermons, prayers, speeches in self-defence and conversations (cf. Lalla and D'Costa 1990: 40) Schumann's dictionaries for Sranan and Saamaka are mostly part of the category recalled since the entries and sentences were reconstructed on the basis of his own knowledge and available field notes. "However, as Schumann checked his texts with his consultants [START_REF] Kramp | Early Creole Lexicography: A Study of C. L. Schumann's Manuscript Dictionnary of Sranan[END_REF]) and the dictionary shows several corrections and insertions that were added later, some parts of the [Sranan] dictionary should be regarded as recorded" (Van den Berg 2007: 19). Some of the nineteenth century West Indian texts are based on recollections of actual (speech) events. George Ross's diary (in D'Costa and Lalla 1989: 20-24), for instance, narrates the occurrences relating to the war and exile of Jamaican Maroons in 1800 and 1801 and includes "recalled conversations." Similarly, we find in Captain Hugh Crow's Memoirs (1830) apparent recollections of speech situations on board of slave ships (D'Costa and Lalla 1989: 24-28). Some of Winer's collection of early vernacular texts from Trinidad, 1839-1851 [START_REF] Winer | Six vernacular texts from Trinidad, 1839-1851[END_REF] are examples of what seems to be recollections of "overheard conversations," as are quite a few of the dialogues in the Spectator texts [START_REF] Winer | Early Trinidadian English Creole: The Spectator Texts[END_REF].

The categories imagined and observed are somewhat difficult to keep apart. It is likely that the author of an imagined dialogue, typical for a work of fiction, would base their embedded dialogue on observed situations or models in order to make the scene more credible and authentic (cf. Mühleisen 2002: 183-207; see Lalla 2005 on the function of Creole in the early phases of Caribbean fiction). The success of such ventriloquism or authenticity depends, however, on the individual's access to real-life models and his or her ability to adopt them. The letters written by the Saamaka Maroons Alabi and Grego and the materials authored by Johannes King should probably be categorized as imagined texts since they were conceived by the authors and/or writers themselves and are not generally based on specific models. Most of the fictional texts fall into this category, as do "imagined dialogues" such as "a conversation on marriage," produced for educational purposes in a religious context in Jamaica (cf. Anon., Wesleyan Methodist Missionary Society, quoted in Brathwaite 2005: 332-36). But the historical language materials from the Caribbean also include texts belonging to the category observed. For instance, van den Berg (2007) notes that there are two language guides for Sranan that each include parts where the authors present expressions that they consider to be typical language use. However, other parts of these guides such as the dialogues and plays that were included for illustrative purposes, are better classified as belonging to the category 'invented' since they are not modelled on a real, observed situation, but were invented based on both some experience and the writers' imagination. Equally, part of the text category 'invented' are the various translations of religious texts into Saamaka such as parts of the Gospel by Wietz in Schuchardt (1914) and into Sranan by Schumann (1781) [START_REF] Arends | The historical study of Creoles and the future of Creole Studies[END_REF]. Text material in planters' diaries, travelogues and letters may be taken to vary between recorded, observed and invented speech.

Technical issues

The encoding of any kind of language material -be it historical or contemporaryin a new and non-standardised form bears all kinds of problems due to the differences between speech and writing. A few issues therefore need to be taken into consideration when interpreting them. One issue relates to whether the mostly non-native speaker authors had sufficient language competence to accurately represent actual language use (Arends 1995b and1995c: 57). While it is likely that second language features entered their writings, they were probably rather limited for at least four reasons (but see also Migge and Van den Berg in press). First, most of the non-native writers appear to have spent a considerable amount of time in the colony before writing in Creole and generally also practiced the language themselves. Second, some of them (e.g. Schumann) worked with native informants who provided part of the data and with whom they also checked the data. Third, comparative studies suggest that the language use represented in the different texts shows important correspondences with chronologically surrounding texts and with modern usage (cf. also Baker and Winer 1999: 117). Finally, differences between texts are not necessarily instances of insufficient competence on the part of the writer, but may equally be due to style, register or variety differences (Arends 1995b;Van den Berg 2007).

However, it is possible that some changes were introduced through the process of writing. For instance, in relation to the relative absence of African vocabulary in historical sources, Stein (1995: 46) argues that (European) writers often excluded Africanisms probably because they did not understand them or did not know how to write them down. In some cases writers "were thinking of their European educated readers, who would neither understand the African elements nor be able to recognize the words" (Stein 1995: 46). Another aspect that may lead to difficulties when assessing documents is variation in orthographical conventions. In the case of most Creole, fully viable orthographies were only devised in the twentieth or twentieth-first century so that most of the written documents from the eighteenth and nineteenth century employed different kinds of writing conventions. Essentially, authors would apply orthographical conventions from the European languages in which they had learned to read and write the Creole. If the writers for the same Creole came from different countries, they would then tend to apply different writing conventions to the same language and therefore obscure similarities in language use (Hinskens and Van Rossem 1995: 76). Moreover, orthographical conventions devised for European languages may have made it difficult to represent forms and distinctions in the Creole because the writing system lacks appropriate graphemes for their representation. Or, certain grammatical differences in the Creole were misrepresented in the texts because non-native speakers of the Creole were not able to perceive them due to influence from their native language and/or lack of linguistic skills (Baker and Winer 1999: 104, 105;Rickford 1987: 81-82). Moreover, depending on their educational and professional background, writers may have had more or less experience in writing in general and thus used a more or less consistent spelling prompting the impression that the writer lacked competence in the language (Bruyn 1995: 41).

It is also likely that some practices underwent stylistic transformation as a result of their being written down. Being used to specific conventions for representing written texts from their native language, writers may have, at least in part, adapted Creole texts, especially those from the text category imagined or invented, to fit a European stylistic mode rather than observe local conventions (cf. Garrett 2000 on the transfer of British style rules in contemporary St. Lucian radio recordings). Baker and Winer (1999: 104) also raise the possibility that authors "might have exaggerated particular features for comic or dramatic effect", however, "they could not stray very far from reality without the risk of losing credibility in the eyes of their readers." A related issue that has hitherto not been examined in much detail concerns the possibility of transfer of grammatical conventions from one Creole to another. Baker and Winer (1999: 104-105), for example, suggest "that visitors often spent time in several different territories and, especially in the Caribbean area where territories are close to one another and speech varieties rather similar, may have been influenced, perhaps unconsciously, by their limited knowledge of one variety in recording speech of another territory". Similar concerns apply to the Surinamese Creoles. Most authors of Saamaka documents had either previously authored Sranan documents (e.g. Schumann) or they had been trained to read and write in Sranan by missionaries using Sranan texts. Even just a cursory look at some of the eighteenth century Saamaka texts (e.g. the Saamaka letters, Wietz in Schuchardt 1914) reveals features that closely resemble Sranan rather than modern Saamaka features. To date it is not clear whether they are transfer features from Sranan or are reflective of diachronic change in Saamaka. Another potentially problematic issue involves the (re)editing of documents. For instance, there appear to be several differences between the original version of Schumann's Saamaka dictionary and the version published in Schuchardt (1914). Finally, persons who wrote texts in place of an author may have also regularized variation.

Linguistic analysis of early texts

The last 25 years or so have witnessed a remarkable growth of interest in early Creole documents mainly due to their greater availability for a number of Creoles. To date the Surinamese Creoles, especially Sranan, and Negerhollands have figured most prominently. However, Baker and Bryun (1999) also edited a collection dealing with eleven texts by Samuel Augustus Mathew in the English Creole of St. Kitts. From the mid-1980s on, [START_REF] Winer | Early Trinidadian English Creole: The Spectator Texts[END_REF][START_REF] Winer | Suzie and Sambo: Linguistic archaism or working class solidarity[END_REF], 1995, 1997, also Winer and Rimmer 1994) has used early Trinidadian literary sources for structural and sociolinguistic analysis and Barbara Lalla and Jean D'Costa (D'Costa and Lalla 1989; Lalla and D'Costa 1990) compiled and analysed a diverse set of Jamaican texts that provide important insights into language use in eighteenth and nineteenth century Jamaica. To date, most of the research has focused on structural linguistic issues. We explore this research in the next section.

Research aims and methodological approaches

Most of the research on historical documents from Caribbean Creoles and Englishes has broadly focused on identifying the linguistic properties of earlier varieties, their etymological origin and their diachronic development. The results have been brought to bear on issues of Creole genesis and their diachronic development such as the relative role of the different sources of Creole grammars, namely substrate and superstrate influence and language internal changes, the exact nature of these sources and the nature of the diffusion of shared features. Some of the research has also dealt with whether contemporary Creoles derive from socalled deep Creoles of from dialectal varieties of a European language (cf. [START_REF] Fields | Early Bajan: Creole or Non-Creole?[END_REF].

Depending on the size of the available corpus, studies have either taken a combined quantitative descriptive approach or a qualitative descriptive approach. In the case of relatively large corpora the former approach is common while the latter approach is usually pursued in the case of smaller data sets.

Overview of grammatically oriented studies of historical Creole documents

Research on historical documents has investigated a number of structural features. Table 1 gives a non-exhaustive list of grammatical areas and Creoles studied. 1 also shows that research on morphosyntactic features including morphological aspects figures prominently while phonological, lexical, syntactic and discourse features have received comparatively little attention. This is reflective of current research trends and the fact that morphosyntactic features are most frequently represented in the data and easily lend themselves to quantitative analysis. The small number of studies on phonological aspects of earlier Creoles may also be due to the special challenges related to the interpretation of individual writers' orthographic choices.

Grammatical area

The findings from research on historical documents written in a Creole

Research on historical documents has provided important insights into the process of Creole formation. It has revealed that contemporary Creole grammars did not emerge abruptly (Bickerton 1981, Thomason andKaufman 1988) but over an extended period of time [START_REF] Arends | Towards a gradualist model of creolization[END_REF]. For instance, [START_REF] Bruyn | Grammaticalization in Creoles: The Development of Determiners and Relative Clauses in Sranan[END_REF] shows that the development including the functional focusing of the singular and plural definite determiners na and den in Sranan stretched over a considerable period of time. In the early texts they functioned as demonstrative determiners as well as definite determiners. However, during the eighteenth and early nineteenth century they came to be more generally used as anaphoric articles (Bruyn 1995: 129) due to grammaticalisation and their demonstrative function was taken over by the elements disi and dati occurring in post-N position. Moreover, the processes of grammaticalisation in Creoles are similar in kind, but maybe faster than those found in other languages.

Another important finding is that Creole genesis was not a single or unique process but involved other processes besides language internal change. For example, Bruyn (1995: 238) argues that "the lack of diachronic development, or rather the relatively sudden establishment of certain functions, can be explained by assuming that interference from substrate languages has played a role." The slaves' first languages also had an impact on features that appear to reflect a diachronic development. While the development of the 'dative' preposition gi from the verb meaning 'to give' seems to have proceeded gradually along accepted universal paths, there is no reason to assume that internal development was solely responsible for this. First, in the case of gi there was no immediate functional need for the development of such a preposition because Sranan had another preposition and also double-object constructions. Second, in the main African substrate languages, the Gbe languages, the main verb meaning 'to give' shows the same kind of functional ambiguity (cf. [START_REF] Migge | Substrate influence in creole formation: the origin of givetype serial verb constructions in the Surinamese plantation creole[END_REF]. This suggests that gi's development is a case of apparent grammaticalization, namely grammaticalization "where the feature does not result from grammaticalization that took place within the Creole language itself but rather from the transfer of the result of a process of grammaticalization that has taken place in another language." (Bruyn 1996: 42).

Sociolinguistic research on early texts

Aims and Methods

Historical texts are not only interesting for the reconstruction of earlier stages of language structure, but also give important insights into language use, language attitudes, communication and the social structure of the early colonies (Roberts 1997: 69-109). The relationship between different social classes and ethnic groups (cf. [START_REF] Brathwaite | The Development of Creole Society in Jamaica 1770-1820[END_REF] in the plantation society is of vital importance for an understanding of the linguistic influences of the groups. Early texts like planters' diaries, letters and other narratives provide observations on, for example, the relationship between Creoles, who were born in the colony, and the newly imported Africans (cf. Roberts 1997: 82-85), but also on the presence of specific European groups like the Scottish. Generally, sociolinguistic research on historical data focuses on the following issues:

 Linguistic and stylistic variation in the texts and what it suggests about the sociolinguistic structure of the language [START_REF] Grant | A note on ethnolectal variation in Mathews' texts[END_REF]Van den Berg 2007, last chapter;Arends 1995b). Here, attitudes towards the variety and the group of speakers it is associated with both through authors and speakers portrayed in dialogues from fictional texts are also at stake [START_REF] Winer | Language varieties in early Trinidadian novels, 1838-1907[END_REF][START_REF] Winer | Early Trinidadian English Creole: The Spectator Texts[END_REF][START_REF] Winer | Six vernacular texts from Trinidad, 1839-1851[END_REF].  Shifts in function and prestige of Caribbean Creoles as a result of their use in writing (cf. Mühleisen 2005a).  Type of interaction between the different groups in the community and patterns of verbal behaviour.  Social information on authors, text type and audience. This also includes information on literacy and literate communication in the plantation society (Roberts 1997: 110-131) as well as the impact of the development of printing on the spread of different types of texts.

4.2 Overview of research in this area [START_REF] Roberts | From Oral to Literate Culture. Colonial Experience in the English West Indies[END_REF] uses a wealth of historical text material to describe the development from oral to literate societies in the Caribbean. [START_REF] Brathwaite | The Development of Creole Society in Jamaica 1770-1820[END_REF] equally draws on early Jamaican texts. The study of variation in an oral language is limited in written documents. As Lalla and D'Costa (1990: 81) point out, "because social attitudes have affected the compilation, survival, or extinction of written records, evidence of historical variation from one century to the next and of social variation at any one period is subtle and elusive." They do, however, examine selected phonological and morpho-syntactic features in the recorded speech of various speakers in early Jamaican texts and correlate these with social factors (level of education, social class, cf. also sample analysis in 4.3).

In [START_REF] Winer | Early Trinidadian English Creole: The Spectator Texts[END_REF] early Trinidadian newspaper articles are analysed with regard to their structural characteristics (lexicon, representation of phonology, morphosyntactic features) and their authenticity value. She uses several internal and external criteria (e.g. consistency, complexity, probability of co-occurrence, intelligibility to audience) to determine a likely correspondence between invented literary dialogues and contemporary real life speech. [START_REF] Winer | Language varieties in early Trinidadian novels, 1838-1907[END_REF] investigate different varieties or styles used in various nineteenth century Trinidadian novels and single out a number of varieties such as Standard British Literary English, British Dialect English, Victorian Gothic English, Victorian Villainous English, Creole Rococo English and Yankee English. The social evaluation of Caribbean Creoles in a historical context is part of the general historical development of discourse about the concept of Creole in [START_REF] Mühleisen | Creole Discourse. Exploring Prestige Formation and Change across Caribbean English-lexicon Creoles[END_REF] and evidence of language attitudes can be found in most early sources written by planters or travellers (e.g. Long, also cited in Lalla and D'Costa 1990: 90;Roberts 1997: 45). [START_REF] Grant | A note on ethnolectal variation in Mathews' texts[END_REF] investigates linguistic variation in Mathews' texts in order to determine the sociolinguistic structure of St. Kitts Creole. He shows that the variable realization of /l/ in intervocalic and word-final position in words like belly, call, tell and kill may be due to phonotactic patterns in the slaves' African languages but argues that "we cannot distinguish characters or their backgrounds on the basis of such variants and must assume that Mathews chose from among the current variants somewhat arbitrarily." (Grant 1999: 127). However, in the case of the definite determiner in the text called 'Mr Thompson' Grant argues that the low occurrence of definite determiners in the speech of the African-born slave is most likely indicative of the existence of an ethnolect while Mathews' use of this features in the same text must be interpreted as a case of linguistic accommodation to that ethnolect. This suggests that in early St. Kitts, at least two and possibly three broad social varieties  European, Creole and African Kittian  could be distinguished. Van den Berg (2007), discussing the linguistic structure of Sranan, shows that some early documents like Schumann's (1783) Sranan dictionary invoke different varieties. Geographical varieties include Djutongo of the Sephardim plantation area, English tongo linked to the plantations of early English planters and Fototongo of the town population. There was also a fourth not explicitly named and "less marked, more neutral variety of plantation Sranan." (p. 380). Some are differentiated by a range of features (e.g. Djutongo vs neutral variety) while others differ by only very few features (e.g. Fototongo vs English tongo). Schumann also invoked two social varieties, Ningretongo, the vernacular of the population of African descent, and Bakratongo, the practices of Europeans and persons of mixed European-African descent. Bakratongo involves greater influence from Dutch while Ningretongo is characterised by greater influence from the first languages of the slaves, namely varieties of Gbe, Kikongo and Akan. There are 36 linguistic variables that have a Ningretongo and/or Bakratongo and/or 'intermediate variant'. Finally, Van den Berg (2007: 385-387) also demonstrates that there is stylistic variation along a formal-informal dimension involving linguistic features such as terms of address, commands and the absence and presence of the past time marker ben, the future marker sa and sentential coordinators and subordinators.

Linguistic-anthropological observations on patterns of verbal and cultural behaviour can also be drawn from early texts. Roberts (1997: 45, 57) cites several historical observations on Afro-Caribbean communicative practices, like for instance, verbal duelling in early nineteenth century market places, or the avoidance of gaze between interactants. Mühleisen (2005c) uses historical sources like planter's diaries to investigate forms of address and their origins in the Caribbean. The development of literate communication in the Caribbean is well documented in Roberts (1997: 110-131).

Sample analysis

Even small data sets can provide interesting insights on the relationship between social structure and speech patterns. One example of an analysis of a single speech event is given in Lalla and D'Costa (1990: 93-96). Here, extensive documentation of a wake held about 1844 for the fisherman Tom Kittle (Murray 1877, reprinted in D'Costa andLalla 1989: 88-111) is used to analyse the speech of the participants represented in this event, "massnegers" (friends and neighbours of the deceased) and landowners from the area. As Lalla and D'Costa 1990: 95 note, "in the data, paradigms such as sex, age, and color can be related to that of social class in terms of educational opportunity. A wide variety of idiolects are indicated and appear to be significantly related to a definable spectrum of variation between J(amaican) C(reole) basilect and acrolect.4 " The following table summarises some of the results (from Lalla and D'Costa 1990: 94). The speech of the participants is not only characterised by the presence or absence of a particular feature, however, but also by the frequency of occurrence. Thus, speaker number 4 (adult woman, uneducated, black) shows the highest frequency of basilectal features, such as fricative replacement, intrusive final vowel or nonverbal predication, not only in comparison with the group of landowners but also in contrast to her male fellow "massneger" (cf. Lalla and D'Costa 1990: 95).

Problems

Conclusions drawn from early written texts on language variation in Caribbean Creoles have to be treated with caution. As noted before, the representation of the language of basilectal speakers depends much on the writer's individual background and attitudes. Furthermore, subtle meaning differences in diverse styles and registers are not always easy to determine, particularly since some features (e.g. prosodic features) inevitably remain unrecorded in writing. Additionally, as Lalla and D'Costa (1990: 98) note in the following quote, diachronic and synchronic variation may not always be easy to keep apart:

Historical reconstruction requires us to resolve an added complexity: the distinction between synchronic from diachronic variants. The corpus does not prove that the variant lects of early J(amaican) C(reole) speakers at any particular stage (e.g. 1790-1810) do or do not represent "interrelated stages of a general change in progress" (Rickford 1981: 47). However, it does establish the existence, from the eighteenth century, of extensive variation (among speakers and within individual usage) in features of basilectal and acrolectal models of Jamaican speech. The data also confirm considerable mixing of such features. By this period, the focusing of speakers on either of these coexistent systems and the mixing of features drawn from them was apparently conditioned by social and economic characteristics and by situational requirements.

Conclusions and outlook

Although the investigation of historical documents written in Caribbean Creoles and Englishes emerged relatively late as a separate area of research (Arends 1995c) it has developed into a thriving area of investigation over the last roughly 25 years. It has so far provided numerous important insights into the emergence and development of Creole grammars and into the social structure of Creole languages and societies, many of which have challenged previous findings reached on the basis of synchronic data alone. However, early language data must not be divorced from other available data types such as information on the social context and the input languages, as well as synchronic language data. Only an analysis that combines all available data types can provide the most comprehensive insights into historical processes (cf. Arends 1995c).

Analyses of historical data have up to now taken diverse foci. However, the majority of studies have to date dealt with structural linguistic issues, investigating the distribution of linguistic features and their implications for Creole genesis and the processes involved. Comparatively few studies are dedicated to exploring sociolinguistic issues. However, these studies have revealed that Creole languages were, from the beginning, complex languages involving a range of clearly distinguishable linguistic varieties. As in the case of other languages, these varieties were linked to existing and emerging local social identities defined by geographical location, social background and ethnicity and to different social situations.

Research on the sociolinguistic structure of early varieties has also led to a reframing of issues such as reliability and the social representativeness of historical data often raised as an issue in previous work (Arends 1995c). More detailed information on the contexts and analyses that attend to both the social and linguistic issues of texts make it possible to classify texts more comprehensively. This in turn will lead to more socially sensitive textual analyses that allow distinguishing between primarily synchronic variation and variation that is indicative of diachronic developments.

There is much room for more sociolinguistic research based on historical written texts in the Caribbean. Ideally, this might be supplemented with pragmatic analyses -an area of analysis that has hardly been touched upon in the Caribbean context. Studies in the relatively recent but thriving field of historical pragmatics (see Jucker 1995) have demonstrated that early written texts can also provide interesting insights on language performances and practices.

1

  For instance, in the Macmillan Caribbean Classics Series: Cassin, Frieda. With Silent Tread. A West Indian Novel (first published Antigua 1890). Edited and with a new introduction by Evelyn O'Callaghan; Jenkins, Edward. Lutchmee and Dilloo. A Study of West Indian Life (first published London 1877). Edited and with a new introduction by David Dabydeen; Orderson, J.W. Creoleana: or Social and Domestic Scenes and Incidents in Barbados in Days of Yore (first published London 1842) and The Fair Barbadian and Faithful Black; or, A Cure for the Gout (first published Liverpool 1835). Edited and with a new introduction by John Gilmore. In the Caribbean Heritage Series of the University of the West Indies Press: Joseph, Edward L. Warner Arundell. The Adventures of a Creole (first published London 1838). Edited by Lise Winer, with annotations and an introduction by Bridget Brereton, Rhonda Cobham, Mary Rimmer, and Lise Winer; Anonymous. Adolphus, a Tale (first published in a serialized version in Trinidadian newspapers in 1853) and Wilkins, Mrs. William Noy. The Slave Son (first published London 1854). Edited by Lise Winer, with annotations and an introduction by Bridget Brereton, Rhonda Cobham, Mary Rimmer, Karen Sánchez-Eppler and Lise Winer; Cobham, Stephen N. Rupert Gray, A Study in Black and White (first published Port of Spain 1907). Edited by Lise Winer, with annotations and an introduction by Bridget Brereton, Rhonda Cobham, Mary Rimmer, and Lise Winer. Warner-Lewis, Warner-

Table 1 :

 1 Grammatical areas investigated in historical documents written in a CreoleTable1demonstrates that most of the research on historical documents has been carried out on the Surinamese Creole Sranan by members of the Amsterdam research group.3 There are also a number of studies on a range of features of early Trinidadian Creole by Winer, of Bajan by Rickford and Handler and Fields, and of St. Kitts Creole English, Jamaican Creole and Saamaka. We expect that more research will in future emerge on the latter Creoles and others because in recent times more historical data have been discovered and are being disseminated. Table

			Language	Reference
	Phonology
	General		Sranan	Smith 1987, Plag and Uffmann 2000;
				Uffmann 2008; Smith 1987;
			Saamaka	Aceto 1996
			St. Kitts and Barbados	Plag 1999
			Jamaican Creole	Lalla and D'Costa 1990: 47-67
			Bajan	Rickford and Handler 1994
			Trinidadian Creole	Winer 1984, 1995
	Vowel system		St. Kitts	Smith 1999
	Morphosyntax
	Copula		Sranan	Arends 1986, 1989, 1995b; Van den
				Berg and Arends 2004a
			Bajan	Rickford and Handler 1994; Fields 1995
	Property items		Sranan	Van den Berg 2007
	Clefting		Sranan	Arends 1986, 1989, 1995b
	Focus-marking		Saamaka	Smith 1996
	Question words	Sranan	Bruyn 1993, Arends 1995b
			Saamaka	Smith 2001
			Atlantic Creoles	Bruyn 1999
			Bajan	Fields 1995
	Serial	Verb	Sranan	Arends 1989, 1995b, Van den Berg 2007
	Constructions			Kramer 2006
			Saamaka	Winer 1984, 1995
			Trinidadian Creole
	TMA		Sranan	Voorhoeve 1957; Van den Berg 2000,
			Saamaka	2001, 2007; Van den Berg and Arends
				2004; Migge and Winford 2009;
			St. Kitts	Corcoran	and	Mufwene	1999;
				Taglimonte 1999
			Bajan	Rickford and Handler 1994; Fields 1995
				Winer 1984, 1995
			Trinidadian Creole
	Verb phrase		Sranan	Van den Berg 2007
	Modifiers		Sranan	Van den Berg 2007
	Complex		Sranan	Bruyn 1995, 1996; Van den Berg 2007
	prepositions			Winer 1984, 1995
			Trinidadian Creole
	Negation		Bajan	Rickford and Handler 1994
			Trinidadian Creole	Winer 1984, 1995
	Pluralization		Bajan	Rickford and Handler 1994
			Trinidadian Creole	Winer 1984, 1995
	Determiner Relativizer	and	Sranan	Arends 1995b; Bruyn 1995a; 1995b; Van den Berg 2007
			Bajan	Fields 1995
	Nouns		Sranan	Bruyn 2007

Selected Basilectal Features in Jamaican Speech at a Single Event, ca. 1850

	Basilectal Features Speakers
		Landowners			"Massnegers"
		1.	2.	3.	4.	5.
		Adult Man	Adult	Adult Man	Adult	Adult Man
		Educated	Woman	Uneducated	Woman	Storyteller
		Color	Uneducated	Brown	Uneducated	Uneducated
		Unknown	Color		Black	Black
			Unknown
	Fricative	-	+	+	+	+
	replacement
	Reduction of	-	+	+	+	+
	consonantal clusters
	Intrusion of final	-	+	+	+	+
	vowel
	Non-English lexeme +	+	+	+	+
	Unmarked pronoun	-	+	+	+	+
	case or tense
	No auxiliary in	-	+	+	+	+
	negative VP
	For/fe complement	-	+	+	+	+
	Nonverbal	-	+	+	+	+
	predication

Table 2 :

 2 Sociolinguistic analysis based on a historical text

Both texts are inArends and Perl (1995: 73-75, 93-242)

Researchers at the Radbound University of Nijmegen and the University of Amsterdam have for quite some time been involved in projects that aim to transcribe and computerize eighteenth century Suriname Creole texts from Sranan Tongo and Saamaka. The latest project is the Suriname Creole Archive project(Van den Berg 2007: 18;[START_REF] Arends | The historical study of Creoles and the future of Creole Studies[END_REF] Arends : 52) completed in 2008. .

Basilect = most removed from Standard English; acrolect = closest to Standard English.

Inhabitants, Air, Water, Diseases, Trade, &c of that Place, with some Relations concerning the Neighbouring Continent, and Islands of America. London: Privately, Privately Printed By ARC for Private Libraries. Smith, Faith. 2002. Creole Recitations: John Jacob Joseph and Colonial Formation in the Late Nineteenth Century Caribbean. Charlottesville and London: University of Virginia Press. Smith, Norval. 1987. The Genesis of the Creole Languages of Surinam. Unpublished dissertation, Universiteit van Amsterdam. Smith, Norval. 1996. 'Focus-marking wE in Saramaccan: Grammaticalization or substrate', in: Changing Meanings, Changing Functions: Peter Baker and Anand Syea (eds) Papers Relating to Grammaticalization in Contact Languages. London: University of Westminster Press, pp. 113-128. Smith, Norval. 1999. 'The vowel system of eighteenth-century St. Kitts Creole: Evidence from the history of the English Creoles?', in: Baker and Bruyn (eds), pp. 145-172. Smith, Norval. 2001. 'Voodoo chile. Differential substrate effects in Saramaccan and Haitian', in: Norval Smith and Tonjes Veenstra (eds) Creolization and Contact. Amsterdam: John Benjamins, pp. 43-81. Stein, Peter. 1995. 'Early creole writing and its effects on the discovery of creole language structures: The case of eighteenth-century Negerhollands', in: Arends (ed), pp. 43-62. Taglimonte, Sali. 1999. 'Modelling an emergent grammar. Past temporal reference in St. Kitts Creole in the 1780s', in: Baker and Bruyn (eds), pp. 201-236. Thomas, John J. 1969 [1869]. The Theory and Practice of Creole Grammar. London: New Beacon Books. Thomason, Sarah G. and Terrence Kaufman 1988. Language Contact, Creolization, and Genetic Linguistics. Berkeley, CA: University of California Press. Uffmann, Christian. 2008. 'Vowel epenthesis and creole syllable structure', in: Susanne Michaelis (ed.) Roots of Creole Structures: Weighing the Contribution of Substrates and Superstrates. Amsterdam: John Benjamins, pp. 123-152. Van den Berg, Margot. 2001. '"Mingo, joù no man": Oud-Sranan in verhoren en verslagen van rechtszaken', Oso 20: 241-253. Van den Berg, Margot. 2000. '"Mi no sal tron tongo": Early Sranan in court records, 1667-1767'. [Unpublished MA thesis, Katholieke Universiteit Nijmegen]. Van den Berg, Margot. 2007. A Grammar of Early Sranan. Durkkerij Manta, Zetten, NL. Van den Berg, Margot and Jacques Arends. 2004a. 'Court records as a source of authentic Early Sranan', in: Geneviève Escure and Armin Schwegler (eds) Creoles, Contact and Language Change: Linguistics and Social Implication. John Benjamins: Amsterdam, pp. 21-34. Van den Berg, Margot and Jacques Arends. 2004b. 'The Saramaka peace treaty in Sranan: An edition of the 1762 text' (including a copy of the original text). Creolica (www.creolica.net) Voorhoeve, Jan. 1957. 'The verbal system of Sranan', Lingua 6: 374-396. Warner-Lewis, Maureen. 1991. Guinea's Other Suns: The African Dynamic in Trinidad Culture. Majority Press.