aa r X i v : . [ phy s i c s . op ti c s ] A p r Optical solitons as quantum objects
Yves Pomeau
Laboratoire de physique statistique de l’Ecole normale sup´erieure,24 Rue Lhomond, 75231 Paris Cedex 05, France.
Martine Le Berre
Laboratoire de photophysique mol´eculaire, Bat.210, 91405 Orsay, France (Dated: November 2, 2018)The intensity of classical bright solitons propagating in linearly coupled identical fibers can bedistributed either in a stable symmetric state at strong coupling or in a stable asymmetric state ifthe coupling is small enough. In the first case, if the initial state is not the equilibrium state, theintensity may switch periodically from fiber to fiber, while in the second case the a-symmetricalstate remains forever, with most of its energy in either fiber. The latter situation makes a state ofpropagation with two exactly reciprocal realizations. In the quantum case, such a situation does notexist as an eigenstate because of the quantum tunneling between the two fibers. Such a tunnelingis a purely quantum phenomenon which does not not exist in the classical theory. We estimate therate of tunneling by quantizing a simplified dynamics derived from the original Lagrangian equationswith test functions. This tunneling could be within reach of the experiments, particularly if thequantum coherence of the soliton can be maintained over a sufficient amount of time.
Lead Paragraph
Usually solitons in optical fibers are assumed to be classical (= non quantum) objectsbecause they are made of a large number of photons. Nevertheless there exist quantumeffects without classical counterpart, like the tunneling under a potential barrier. Weinvestigate one possible realization of such a quantum tunneling with solitons as basicentities. Specifically, we consider a soliton propagating in two linearly coupled fibersthat are assumed identical. It has been known for some time that, at small enoughcoupling, asymmetric solitons only can propagate and be stable. The amplitude ofsuch asymmetric solitons is predominantly in either fiber and remains there foreverclassically. This makes, for a given energy, two possible steady states exactly symmet-rical with respect to each other under permutation of the two fibers. In the quantumversion of the same problem, the two solitons merge into a single quantum state shar-ing a quantum amplitude spread between the two fibers, because of the possibility of quantum tunneling from one fiber to the other. We study this problem thanks to areduced set of equations derived from the full set of coupled nonlinear PDE’s by choos-ing convenient trial functions for the classical soliton dynamics. Thanks to this choice,the bifurcation pattern of the soliton solution in the coupled fibers is well reproduced.Because the trial set generates dynamical equations with a Lagrange structure, thisLagrangian system is relatively easy to quantize. To obtain the quantum amplitude oftransmission by tunneling under the barrier, one replaces the original Hamiltonian sys-tem by its Euclidean counterpart. Orders of magnitude relevant for a possible physicalapplication are given.
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I. INTRODUCTION
Generally speaking a soliton is a localized solution of an equation for the propagation of a field envelope. It stayslocalized under the opposite effects of linear dispersion tending to spread the wave and of nonlinearity making thewave steeper. We make one more step by considering this soliton as a ‘true’ particle, that is by seeing it as a classicalobject that should be ultimately quantized to keep the consistency of our view of the physical world. This is of coursenot a new idea, see for instance the review [1] on the quantization of various nonlinear equations for classical fields.Quantum effects are irrelevant for macroscopic phenomena like solitary waves in a water channel. However, thereis an instance of solitonic physics where quantization could bring significant new effects, namely the propagationof optical solitons in fibers: there the amplitude of the wave may be small enough to yield solitons with not toolarge action, measured in units of Planck’s constant h π . If the action is much larger than this quantum unit one isin the classical regime, many trajectories contribute to the saddle point of the Feynman integral [2] and quantuminterferences between coherent quantum states become practically impossible. Conversely, if the action is not too largecompared to this quantum unit, one could observe quantum phenomena as tunneling and interferences. Moreover,even if the quantum state has a coherence time shorter than the tunneling time, there is still quantum tunneling,but at a reduced rate because the build-up of the state on the other side of the barrier is slowed down [3]. Belowwe show that quantum tunneling of a soliton may occur between two weakly coupled fibers, and we discuss thepossibility of quantum interference between the two states carried by each fiber. The starting point of our study isthe well established result that classically a soliton injected in a given fiber cannot switch to the other fiber, whenthe coupling is less than a certain critical value. In this range the soliton evolves towards the stable a-symmetricsolution having its energy predominantly in the initial fiber [4]. There exist another stable solution which is obtainedby permutation of the two fibers. At small coupling the two a-symmetrical states are separated by a finite barrierthat cannot be crossed classically. We predict that this may be wrong in practice because of quantum tunneling.Our derivation is based on the calculation of the tunneling probability which writes in the WKB approximation, as T = exp ( − S/ ~ ), where S is the physical action associated to tunneling [5]. The quantity S/ ~ may be small, evenfor a pulse with a large number of photons, because all dynamical phenomena we consider, like the balance betweenthe nonlinearity, the group velocity dispersion and the coupling, imply small perturbations to the dominant effectresulting from the linear dispersionless terms of Maxwell’s equations. The perturbations we consider are as small asthe nonlinear term n I with respect to the dominant term n , in the expansion of the refractive index n = n + n I for the Kerr medium of the fiber, I being the optical intensity. In this sense a soliton is a bound state of photons:photons are attracted to each other by the focusing nonlinearity. The soliton resembles the atom of a heavy elementwhich is a quantum object made of many electrons and nucleons. Atomic physics has also to do with energies muchsmaller than the rest energy of the particles (electrons and nucleons) making the atom. Another idea of atomic (andquantum!) physics is relevant for our goal: If the atom remains in its ground state, it is able to make interferenceswith a wavenumber depending on its mass and velocity only, independently of the details of the state of its electronsand nucleons. In our study we assume the absorption and change of frequency of the photons by inelastic and/orRaman scattering to be negligible, and discuss the role of these effects in interference experiments in the last section.Let us outline the organization of this paper. In the Section II we introduce the classical model of propagation ofsolitons in fibers. First we deal with the single fiber, then with the two coupled fibers. There is nothing new here andwe focus on what is relevant for us, namely the bifurcation diagram as a function of the coupling. For strong couplingthe stable solution is symmetric (with intensity equally shared between the two fibers). Below a critical coupling, theclassical prediction is that the stable soliton is a-symmetric, with a high amplitude in one fiber and a small amplitudein the other, as said above. The details of the transition are a bit complex because it is subcritical. This has beenstudied [4] by direct numerical solution of the coupled PDE’s describing this problem (equations (6) and (7) below).However interesting it is, this model presents some difficulties for our quantization problem. Therefore, in the nextsection III we outline another approach to the same problem, namely we use the Lagrange formalism to computeapproximate solutions with trial functions (instead of the full unknown solution). Thanks to an approriate choice ofthese functions, the pattern of bifurcations of the asymmetric to symmetric solitons, known from the direct numericalsimulations, is well recovered. We use the same trial functions as Malomed et al. [6] who discuss very thoroughly thegeneral issue in a paper that we recommend to the interested reader.As explained in section IV, this ‘trial dynamics’ is used then to quantize the system. This is of course not exact,but requires far less formalism than the full quantization of the two coupled nonlinear field equations. Thanks to thismethod, one can use standard results and methods of quantum mechanics for systems with a few degrees of freedom(as opposed to field theories). In particular, we can compute the trajectory under the potential barrier, found bymultiplying the propagation variable z, and the Hamilton-Jacobi action S by i . This yields a well defined problemof Hamiltonian mechanics, called sometimes the Euclidean version of the initial problem. The tunneling factor isderived from the action of the heteroclinic trajectory joining the two equilibria: a stable equilibrium in the originalHamiltonian system remains an equilibrium in the Euclidean one, but it becomes unstable there. It turns out thatthe tunneling probability depends algebraically on the coupling between the two fibers. This is a significant remarkfor possible applications, because it yields a much smoother dependence with respect to the coupling than the usualexponentially small tunneling amplitudes.The last section summarizes the main results of this paper, discusses the possibility of interferences and presentssome ideas for possible applications. Quantitative predictions rest on the rather complex problem of turning backfrom the dimensionless equations used throughout this work to quantities with a physical dimension, something donein the Appendix. II. CLASSICAL PROPAGATION OF SOLITONS IN COUPLED FIBERS: THE GENERAL MODEL
The mathematical model for the dissipationless propagation of optical solitons in one fiber is the classical (= nonquantum) nonlinear Schr¨odinger equation: i ∂E∂z + α ∂ E∂t + β | E | E = 0 . (1)Even though this equation is called nonlinear Schr¨odinger (NLS), it does not mean at all that it makes a quantumsystem. It only resembles the usual Schr¨odinger equation, but it describes a purely classical field, exactly as Maxwell’sequations do for an EM field. This equation is written with real coefficients α and β carrying a physical dimensionto make possible the discussion (see Appendix) of the order of magnitude of the physical effects to be expected. Thefield E is the complex amplitude of the electric field in the wave. We take it as a scalar, although polarization effectscould be brought into the picture in principle. This equation is derived in the Fresnel approximation, assuming thatthe changes of amplitude along the fiber are much slower and on much longer scales than the oscillations of the opticalfield itself and it is also written in the frame of reference moving with the speed of the envelop of the wave, wherethe position variable is z . For α and β real, this equation has a Lagrange-like structure. It cancels the first ordervariation of the ‘action’ S = Z dz Z d t (cid:20) i (cid:18) E ∂E∂z − E ∂E∂z (cid:19) − α | ∂E∂t | + β | E | (cid:21) . (2)In this equation E is the complex conjugate of E . The writing of the action in equation (2) brings in an importantproblem, because it is not ‘the’ physical action. Such a physical action has to have the dimension of the product ofan energy and of a time. Therefore, the action written in equation (2) cannot be an action from the point of viewof physical dimensions. The physical action of the EM field is proportional to S , its derivation is postponed to theAppendix. An overall constant multiplying factor does not change the Euler-Lagrange equations, but it is crucialwhen quantizing this system because this relies on a comparison between the action and ~ , two quantities with thesame physical dimension.By rescaling E → uβ − / (assuming β positive to be in the focusing case where solitons exist), and t → t (2 α ) / ,one obtains the dimensionless nonlinear Shr¨odinger equation: i ∂u∂z + 12 ∂ u∂t + | u | u = 0 . (3)This equation has a number of interesting symmetries. In addition to the Galilean invariance (if u ( z, t ) is a solution,then u ( z, t − z/C ) e iC ( t − z C ) is also a solution), it has a dilation symmmetry: if u ( z, t ) is a solution and µ an arbitraryreal number, then µ u ( zµ , tµ ) is also a solution. It has a two parameters family of soliton solutions: u = νe iϕ cosh( νt ) = νe iϕ sech( νt ) . (4)In this solution, ν is any real number and the phase ϕ is ϕ = ν z + ϕ , with ϕ arbitrary constant phase.Note that z plays here the same role as the time in the usual Schr¨odinger equation. Among the conserved quantitiesassociated to any solution of the NLS equation, let us write the ”energy” H = 12 Z d t (cid:20) | ∂u∂t | − | u | (cid:21) . (5)Suppose now that, instead of a single optical fiber, we have two identical coupled fibers, and that the coupling islinear and preserves the symmetry between the fibers. The propagation of solitons in this system has been studied inthe last fifteen years [4], [6]-[10].To describe the two coupled fibers supporting solitons we introduce two focusing NLS equations, written in adimensionless form: i ∂u∂z + 12 ∂ u∂t + | u | u = − κv, (6)and i ∂v∂z + 12 ∂ v∂t + | v | v = − κu, (7)where κ is the strength of the linear coupling, and we define the ”mass” Q = Z ( | u | + | v | ) dt. (8)which is a constant of motion. Consider solutions of the form u ( z, t ) = U ( t ) e iqz and v ( z, t ) = V ( t ) e iqz . Because ofthe common phase factor e iqz the z -dependence cancels out and the two functions U ( t ) and V ( t ) are solutions of thetwo coupled ordinary differential equations: ( − qU +
12 d U d t + | U | U = − κV − qV +
12 d V d t + | V | V = − κU. (9)For the solution to decrease to zero when t tends to plus or minus infinity one must have q >
0. Furthermore thesign of κ can be changed by changing U into − U for instance and keeping V the same. We choose κ positive thatcorresponds to in-phase stationary solutions ( U, V ), the out of phase ones being unstable [4].This set of equations has been studied numerically and analytically [4]. An exact calculation shows that thesymmetric solution U = V always exists and is linearly stable in the range Q √ κ ≤ √ . For higher values of thisratio, the symmetric solution looses its stability and an a-symmetric solution branches off. While it is not explicitelymentioned in [4], the subcritical character of the bifurcation can be deduced from Fig.11 of the paper by Akhmedievand Soto-Crespo(1994) when using Q √ κ as control parameter and qκ as order parameter, i.e. by rotating the figure.Consequently no stable and weakly asymmetric solutions branches off the unstable symmetric soliton for κ slightysmaller than the onset of linear stability, although an unstable asymmetric solution branches off at values of κ slightlylarger than the critical one. Furthermore a branch of stable asymmetric solitons goes continuously from κ = 0 to afinite coupling, slightly larger than the value of linear instability of the symmetric soliton. The stable asymmetricsoliton disappears by a saddle-node bifurcation for a value of the coupling that is, by a numerical coincidence, veryclose to but smaller than the onset of linear stability of the symmetric soliton. At this saddle-node bifurcation theunstable and stable a-symmetric solutions merge to disappear at smaller values of Q √ κ .In the numerical investigations of this problem an interesting phenomenon comes into play, namely the radiation ofenergy at large distances of the solitons. The amount of radiation is stronger when the initial conditions are furtheraway from a stable solution [4]. This radiation happens in the far wings of the time dependent amplitude profiles( | u ( t ) | , | v ( t ) | ), where the full equation reduces to its linear part. Although very strongly dispersive this describesradiation by wave packets of ever increasing width, but carrying nevertheless energy and eventually other invariantsto infinity. Such a coupling between a localized system and the infinitely many degrees of freedom of a radiatingfield may lead to irreversible effects [11]. It shows how subtle may be the distinction between ‘dissipative’ and‘nondissipative’ systems as soon as one goes beyond the obvious. Irreversible process due to radiation may not evenrequire an infinitely extended physical space. They may also take place in the reciprocal (or momentum) space bycascade of energy toward smaller and smaller scales, a typically nonlinear phenomenon [12]. We plan to come back tothe issue of the effect of radiation on quantum phenomena in the present problem. We shall neglect this kind of effectin the following, since they cannot be taken into account within our simple formalism. Even though the radiativelosses are present, it was shown by Fadeev and Korepin [1] that they do not destroy the solitons in a single fiber,when they are included in the quantized version of the NLS equation.In the coming section we shall derive a reduced set of equations describing the propagation of soliton in coupledfibers. Indeed this reduction from the original PDE’s to a set of coupled ODE’s cannot be quantitatively exact.However with the same choice of trial functions as Malomed et al.[6] we obtain at least a reduced system with theright pattern of bifurcation at decreasing coupling. The fundamental interest of this reduction is that it allows us toquantize the dynamical system rather straightforwardly. III. CLASSICAL PROPAGATION OF SOLITONS IN COUPLED FIBERS: THE REDUCED DYNAMICS
Because of the lack of analytical solution in general, we follow an idea used already by various authors, that allowsto understand in a fairly detailed way the results of the direct numerical simulation by using an analytical approach.This follows the general method of research of extrema of functionals by trial functions: dynamics can be reducedto a minimization problem, then one restricts the function space where this minimization is done to a space of trialfunctions depending explicitly on a few parameters and one studies the dynamical properties in this reduced space.Since we know the results of the direct numerical simulations it is in principle possible to check the quality of theapproximation by comparing its predictions and the ‘exact’ results. This is necessary because the method of trialfunctions does not rely on a small or large parameter and so cannot hope to be ‘exact’ or close to exact in the usualmathematical meaning of the word. The papers by Malomed et al. [6] discuss in depth the choice of the trial functions.We shall not reproduce this discussion here where we take their set of ‘optimized’ trial functions, following as muchas possible their notations.The starting point is the writing of the action for the coupled NLS equations: S NLS = Z z z d z Z d t (cid:20) i (cid:18) u ∂u∂z − u ∂u∂z (cid:19) + i (cid:18) v ∂v∂z − v ∂v∂z (cid:19) − | ∂u∂t | + 12 | u | − | ∂v∂t | + 12 | v | + κ ( uv + vu ) (cid:21) . (10)As it can be checked the action S NLS is proportional to ( z − z ) whenever the functions u ( z, t ) and v ( z, t ) arestationary solutions (with respect to the variable z ) of the two coupled NLS equations (6) and (7) or functionsproportional to the same phase factor e iqz . The problem we consider now is how does the coupling change thepropagation of solitons. For that purpose we reduce the dependence with respect to t to an imposed form (the trialfunction) with arbitrary z -dependent coefficients, the trial parameters. Putting this trial form into the action integraland performing the integration over the variable t yields a functional of the parameters of the trial function that arethemselves functions of z . Doing now the variation with respect to those functions, one finds at the end a set ofODE’s for functions of z only.The choice of the trial functions is inspired by the soliton solution in a single fiber and it respects the symmetrybetween the two fibers. Following Uzunov et al.[6] one takes: u ( z, t ) = a ( z ) p η ( z )sech [ η ( z ) t ] cos(Θ( z )) exp (cid:2) i (cid:0) Φ( z ) + Ψ( z ) + q ( z ) t (cid:1)(cid:3) , (11)and v ( z, t ) = a ( z ) p η ( z )sech [ η ( z ) t ] sin(Θ( z )) exp (cid:2) i (cid:0) Φ( z ) − Ψ( z ) + q ( z ) t (cid:1)(cid:3) . (12)In the case of a single fiber carrying a soliton of amplitude u , the trial function u ( z, t ) becomes the exact one-solitonsolution with a = √ η , Θ = q = 0, Φ = zη / u and v . The angle Θ could be replaced by anotherparameter in another trial function, not necessarily a circular function. Inserting this trial form into the action S NLS and performing the integration over t , which is possible because the dependence with respect to t is fully explicit inthe trial functions, one finds a reduced action that is itself the integral over z of the Lagrange function: L = 2 κa ( z ) cos(2Ψ) sin(2Θ) − a cos(2Θ) dΨd z − a η sin (2Θ) + 23 a η − a η − a dΦd z − a π η (cid:18) d q d z + 2 q (cid:19) . (13)Up to obvious change in notations (from our z to ζ , from κ to K , etc.) this Lagrange function is identical to the onewritten by Uzunov et al. [6] but for a misprint in their paper where the term q in the last parentheses became q without harming the rest of their calculation. The parameters of the trial function are five functions of z : a , Θ, Ψ, η and q . The equations of motion for those five functions are derived by variation of the action, namely the integralover z of L . They read: d a d z = 0 , (14)derived by variation with respect to Φ, and dΘd z = − κ sin(2Ψ)sin(2Θ) dΨd z = a η sin(2Θ) cos(2Θ) − κ cos(2Ψ) cos(2Θ) d η d z = − qη d q d z = − q + π (cid:2) η − a η (cid:0) − sin (2Θ) (cid:1)(cid:3) (15)derived by variation with respect to Ψ , Θ , q , and η respectivelyThe parameter a can be absorbed in the redefinition of κ and will be set to 1 below (that corresponds to a mass Q = 2).The soliton solutions are z -independent solutions of this set of equations. There are two classes of soliton solutionsin this model, depending on the coupling parameter. For any coupling there exists a symmetric soliton, with equalintensity in both fibers, i.e. Θ = π . At small coupling this symmetric solution is unstable against asymmetric soliton. -1 -0.5 0.5 1 x (cid:144) κ FIG. 1: Inverse of coupling coefficient κ − versus the stationary values of x = cos (2Θ). For large coupling, i.e. κ > /
6, thesymmetric solution, x = 0, is the only one stable. Such an asymmetric soliton is found by canceling the z -derivatives in equations (15) and choosing cos(2Ψ) = 1. Thisyields the relation between the coupling coefficient κ and the balance parameter for the intensity x = cos 2Θ κ − = 6(1 + x ) √ − x , (16)which is illustrated in Fig.1. Using the trial functions, the bifurcation in the set of possible solutions is found tooccur at the critical value, κ c = 1 / κ c = √ / ≤ κ ≤ √ (0 . ≤ κ ≤ . a = 1 it becomes: H cltrial = − κ cos(2Ψ) sin(2Θ) + 13 η sin (2Θ) − η η π q η . (17)Note that the Lagrangian (13) includes terms linear with respect to first derivatives (with respect to z ). It meansthat the two successive operations of choosing trial functions and averaging over the retarded time t , lead from theLagrangian formalism to the Hamiltonian one, with S = Z ( X i =1 , p i dq i − H dz ) (18)The first term in the r.h.s. of equation (18) will be the one responsible for the Euclidian action derived in the nextsection. As already noticed by Uzunov et al., equations (14) to (15) are the Hamilton equations of a two-degrees offreedom system. The two pairs of conjugate variables are { Ψ , x = 2 cos(2Θ) } , and { q, y = π η } , i.e. the phaseand amplitude differences, as well as the chirp and width, respectively. We are interested in the value of H forsteady states, that turns out to be a simple function of the coupling κ . That should give an idea of how the energychanges when the variables are different of their values in the steady state(s). In order to preserve the connectionwith a potential energy in the usual sense we impose that, at the equilibrium points, this ‘potential’ energy is at anextremum. This is realized (probably not uniquely) by plugging into H the values of q and Ψ at the various equilibria,that is Ψ = 0 and q = 0 to cancel the conjugate momenta. This yields: H pot = − κ sin(2Θ) + 13 η sin (2Θ) − η η . (19)This ‘potential’ energy depends on two parameters, x = cos(2Θ) and η , and it is plotted in Fig. 2 for variouscoupling strength to show the bifurcation of the equilibria from a single equilibrium at large coupling, Fig. 2( a ), to (a) -0.75 -0.5 -0.25 0 0.25 0.5 0.750.20.40.60.81 x Η (b) -0.6 -0.4 -0.2 0 0.2 0.4 0.60.450.50.550.60.650.70.75 x Η (c) - - FIG. 2: Level lines of the potential in the plane ( x, η ), (a) for κ = 0 .
5, only the symmetric solution x = 0 is stable, (b)for κ = 0 .
18 which belongs to the subcritical domain in Fig.1 where symmetric and a-symmetric solutions are stable, (c) for κ = 0 . a more complex pattern, as the coupling decreases. In particular, some sort of barrier is evident in Fig. 2( c ). Itseparates the two deep minima of the potential lying each in the vicinity of ( ± , , /
2) is a saddle point of thepotential energy.The above picture illustrates the known results: classically there is no way for a soliton initially in a given fiber, toescape through the other fiber, at low coupling, because of the barrier. Before to present the quantum version of thisproblem, let us precise what is the low coupling range in terms of physical quantities. Note first that the low-couplingrange writes a κ > , (20)for an incident soliton of the form u (0 , t ) = a/ch ( at ) injected in one of the two fibers ( a = 1 above). Secondlylet us define the scaling quantities in equations(6)-(7), by using the soliton units, z = z phys /L D , t = t phys /τ , u, v = q πn ′ λ L D B , , and κ = κ phys L D , where L D = τ k ” is the dispersion length, and B , are the slowly varyingamplitudes of the electric field (see appendix). The relation (20) becomes n I M = n ′ (cid:12)(cid:12) B M (cid:12)(cid:12) > π λ κ phys , (21)or, n I M > λ L c when introducing the switching length L c = π κ phys defined for the CW linear regime. Using therelation (A3), the low coupling range also writes L c > πL D . (22) IV. SEMICLASSICAL QUANTIZATION OF THE COUPLED FIBER SYSTEM
Before computing the quantum tunneling, let us recall the main differences between the classical solitonic solutionand its quantized form. In quantum mechanics a state localized on one side or on the other only is not an eigenstateof the system, because of the possibility of tunneling. Therefore if one starts at ‘time’ zero with all the amplitude onone side (meaning all the probability in one of the two possible asymmetric states), after the time of tunneling thiswill be transferred to the other side and eventually oscillate between the two sides. It is also possible to inject at theinput of the dual core fiber, the quantum ground state, which is symmetrical, with equal amplitude in the two sides.To have a physical image of the process by which the transition occurs between the two asymmetric states, one mayrecall that the number of photons is not fixed in the quantized soliton, so that it fluctuates in both fibers. Therefore,the fluctuations may bring one fiber into the soliton state, although the other goes to the state without soliton, andthe two states switch in the course of time, as studied below.Let us now outline how to compute the quantum tunneling between the two fibers. Because we have a classical field,the quantization of the coupled equations (6), (7) for the two fibers belongs to the general problem of quantizationof field theories. Although this may be done formally, it requires a rather heavy machinery in any case. Fortunatelythere are various possible short cuts in this derivation. The most obvious one is to reduce PDE’s system to a set ofODE’s, by using trial functions depending on a certain set of unknown parameters. By refining the choice of trialfunctions ad infinitum, namely by introducing trial function with more and more parameters, one should converge inprinciple toward the exact result. But we will merely use the above described trial functions. The Euler–Lagrangecondition of stationarity of the action yields a set of dynamical (in ‘time’ z ) equation, that can be formally quantizedbecause it has a symplectic structure.This is what we are going to do, except for one point. It is possible to short cut all this explicit quantization in theWKB limit, where the wave function is expressed by means of the classical Hamilton-Jacobi action, Φ = A exp( i S / ~ ).This is the well-known quasi–classical limit, that restricts oneself to situations where any action involved is typicallymuch bigger than ~ . This seems a reasonable limit, but it does not necessarily cover all possible situations-we shallcome at the end to what seems to be ‘the’ standard experimental situation in this respect. The WKB limit is especiallyconvenient for treating tunneling problems, because it amounts to calculate the imaginary part of the action (whichis complex) and to put at the end ~ at the right place. Indeed the tunneling factor is is given by T = exp ( − S E / ~ ),at leading order. Here S E is the imaginary part of the action, which enters then in the modulus of the wave functionas a real exponent (instead of the usual imaginary exponent relevant for the classical limit of quantum mechanics).This imaginary part of the action is calculated by two steps. First one has to change the conjugate variables ( q, p )into ( q, ip ) in the classical Hamilton-Jacobi formulation of quantum mechanics, the Hamiltonian H ( q, p ) becoming H ( q, ip ). Secondly one is left with a problem of extremalization of a new action, the Euclidean action, that is formallyanother problem of classical mechanics. For instance in the often presented problem of a particle of energy E in adouble well potential V ( q ), with Hamiltonian H = p m + V ( q ), the Euclidean action is calculated with the abbreviatedaction [13] S E = Z q [ z f ] q [0] pdq (23). derived from the Hamiltonian H E = − p m + V ( q ) but with the same energy as the one of the classical motion. Forthe potential this means that it gets rotated by 180 degrees, thus exhibiting two ”hills” of maximal energy. The valuesof q [0] and q [ z f ] in equation (23) are those of the classical turning points defined by E = V ( q ) . To calculate S E , onehas to find a trajectory joining these points, namely to calculate an Euclidean path integral. This is performed bysolving the Hamilton equations for the Euclidean Hamiltonian ∂q∂z = ∂H E ∂p∂p∂z = − ∂H E ∂q (24)by taking as initial conditions, the known value q [0], and an unknown value p [0]. By varying the latter value, onefinally converges towards a trajectory ending at q [ z f ], which provides the action defined in equation (23). Note thatequations (24) are obtained from the classical Hamiltonian system ( which is identical to equation (24) but with H inplace of H E ), by changing z in iz , and p in ip . The change to an imaginary ”time” ( from z to iz here) amounts togo from a Minkowskian to an Euclidean metric. Therefore equations (24) are called ”Euclidean equations of motion”,and their classical solution joining the two ”vacua” of the double-well potential, often named ”kink solution”, is anexample of an instanton [14] in quantum mechanics.In the above example the variables (p, q) are the impulsion and position of a particle in a 1D potential. General-ization to cases of a multidimensional set of generalized coordinates and momenta leads to similar relations [13]. A. Semi-classical Action.
To put all those principles in practice we have to formalize the dynamical system (equation (15)) in terms ofcanonically conjugate variables. Once this is done, the Euclidean equations of motion are found by multiplying the”time” z and the momenta by i . As noted in section III, the reduced equations (15) are those of an Hamiltoniansystem with two degrees of freedom, therefore a simple choice for conjugate variables ( q j , p j ) with j = 1 ,
2, is to takethe pair ( x = cos (2Θ) ,y = π η ) as coordinates and (2Ψ ,q ) as their conjugate momenta. The Euclidean Hamiltonianis obtained from the classical one in equation (19), by changing cos(2Ψ) into cosh(2Ψ), and q into − q . It becomes H E , trial = − κ cosh(2Ψ) sin(2Θ) + 13 η sin (2Θ) − η η − π q η . (25)The semi-classical dynamics is then driven by the new set of four (Euclidean) equations, that are the Hamiltonequations for the conjugate variables ( q j , p j ), deduced from the Euclidean Hamiltonian (25) d x d z = − κ sinh(2Ψ) √ − x z = ηx − κ cosh(2Ψ) x √ − x d η d z = 2 qη d q d z = 2 q + π ( η − η x ) (26) -1 -0.5 0 0.5 10.50.60.70.80.91 x η FIG. 3: Quantum trajectory superposed to the potential for κ = 0 .
1: only two asymmetric solutions are stable.
As in the case of a particle in a double well potential, calculating the probability for the soliton to tunnel througha classically forbidden region ( H pot ) with the Minskowskian space path integral, corresponds to calculating thetransition probability to tunnel through a classically allowed region ( − H pot ) in the Euclidean path integral, withthe action S E = Z q R q L ( p dq + p dq ) (27). where q L , q R are the coordinates of the turning points. To perform the integration giving the action, it is enough tochoose a convenient integration path in the Euclidean plane connecting the two minima M ( x M , η M ) of the classicalpotential in Fig. 3, which become the maxima of the Euclidean potential. For small values of κ , one has H pot , M ≈− (1 + 18 κ ). In the present case it is easier to carry the integral from x = 0 up to x M . We set the value of theHamiltonian H close to H pot , M . Because of the symmetry of the heteroclinic trajectory joining the two extrema ofthe potential, we have to choose the initial condition q = 0. Then the initial value of the phase difference Ψ isdeduced from equation (25), and we have only one initial parameter to adjust, η , in order that the trajectory endswith a vanishing impulse q f = Ψ f = 0 close to the extrema M ( x M , η M ) in the plane ( x, η ). The integration path isshown in Fig.3. The action along the semi-classical trajectory is given by the expression (27) that writes with ournotations S E = 2 (cid:12)(cid:12)(cid:12)(cid:12)Z z = z f z =0 (cid:20) − z ) κ sinh(2Ψ( z )) p − x ( z ) − π q ( z ) η ( z ) (cid:21) d z (cid:12)(cid:12)(cid:12)(cid:12) . (28)The numerical result of the integration is shown in Fig. 4 which displays the action as a function of the couplingparameter κ in a logarithmic scale. In the domain of existence of the asymmetric solution, κ − > Log (E / κ ) (Action/2) ( φ ) ( η ) FIG. 4: Action/2 (squares), Maximum of the impulse phase Φ (circles), and η (triangles) versus the scaled coupling parameter S E = 2 ln( κ c /κ ) (29)that holds true with a precision better than 1 per cent over many decades, with the numerical value κ c = 0 . .
18. Note that the relation (29) holds true except in the close vicinity ofthe bifurcation point, not visible in Fig. 4. The ln -dependence in equation (29) follows straightforwardly from thesubstitution of exponentials for the hyperbolic sine in the equation of motion for Euclidean dynamics. We also reportin Fig. 4 the dependence of Ψ and η as function of κ . At x = 0 the solution becomes transiently symmetric,sin(2Θ) = 0, but its width is different from the symmetric value, η = , and the impulse Ψ is maximum. We showthat Ψ evolves very much as the action, while η is quite constant. Actually the heteroclinic trajectory drawn inFig.(3) passes through the abscissa x = 0 approximately at the ordinate η ∼ .
67 whatever the value of the couplingconstant, while the impulse here increases like ln( κ c /κ ). This result shows the leading role of the conjugate variablesΨ and x = cos(2Θ) in the dynamics. At this stage it is interesting to compare the latter result (29) with the actionderived by a simpler choice of trial functions, based on the hypothesis of constant width soliton (and of no chirp),as proposed by Par´e [8] and Kivshar [9]. In these simpler cases, one obtains a single degree of freedom Hamiltoniandynamics. The approximate calculation of the Euclidean action may be done analytically, and leads to similar resultsin both cases. With the notations of Kivshar, for example, using as conjugate variables (Φ , ∆), the calculation of theaction amounts to carry the integral S E = R +1 − Φ(∆)d∆, the function Φ(∆) being given explicitly in [9]. In the limitof a small coupling and with κ = γ − , the equation for Φ reduces, at leading order, to Φ ≈ i ln( γ ) + ˆΦ where ˆΦ is thesolution of I (∆) = e i ˆΦ that is of order 1. Therefore in this limit γ large (equivalent to small coupling), Φ ≈ i ln( γ ) sothat the action associated to tunneling is just S ≈ i ln( γγ c ), where γ c is a constant.Summarizing the Euclidean action obeys the law (29) in all cases of trial functions we have considered, i.e. for asingle degree of freedom Hamiltonian as well as with two degrees of freedom. Consequently, it does not seem necessaryto refine more our model to obtain the information we need, i.e. the order of magnitude of the tunneling amplitude. B. Tunneling factor.
The possibility for the soliton to tunnel from one fiber to the other in real space, is measured by the transmissioncoefficient, with the expression T = (cid:12)(cid:12) FA (cid:12)(cid:12) in a double well tunneling problem, with F, A the amplitudes of thetransmitted and incident waves, respectively [5] . It has already been noted that the transmission is given by T = exp ( − S/ ~ ) at leading order. In practice S is the ”physical action”, having the same dimension as ~ . Therefore, tocalculate the ”true” transmission for the soliton in the two coupled fibers one has to multiply the dimensionless action S E by an appropriate coefficient s (1) depending on the properties of the fiber and of the characteristics of the EMwave, this giving lastly the ”physical action” S physE = s (1) S E which has the dimension of ~ . As shown in the appendix2 s (1) / ~ = γ/ ( ω τ ) (30)1where γ ∼ σε c k ” / ( n ′ ~ ) . (31)The value of γ depends on the fiber parameters n ′ and σ , cross section of the fiber. Let us consider 10 µm areasilica fibers with n ′ = 2 . − ( m/V ) as given in ([16]), ([15]). With the values of coefficients given in the appendixin MKS units, the coefficient γ is about 3 . .With equations (29)-(31), the transmission coefficient T = exp − [ 2 γ ( ω τ ) ln ( κ c /κ )] , (32)or T = ( κκ c ) γ ( ω τ , (33)behaves as a power law, that is smoother than the usual exponential in tunneling amplitudes. The tunneling ispossible when the exponent in equation (32) is ”not too big”. In the semi-classical regime considered above, thephase of the wave-function is derived by expansion at lowest order with respect to ~ . This requires that the exponentln( T ) = γ ( ω τ ) ln ( κ c /κ ) is much larger than unity, then the probability of tunneling is obviously weak. When theexponent becomes smaller or of order unity, one is in the ”pure quantum limit”, and the previous derivation is nomore valid, since the wave-function cannot reduce to its first order term in ~ . Nevertheless we can assert by continuityargument that tunneling continue to exist, and that it is likely much more efficient. The boundary between these twolimits can be defined by γ ( ω τ ) ln ( κ c /κ ) = 1 . (34)This dependence is drawn in Fig. 5 for the value of γ given above. The quantum regime is reached as soon as thepulse duration is longer than a ps . Therefore quantum tunneling seems within reach of present days experiments.
20 40 60 80 100 κ c (cid:144) κ τ H fs L FIG. 5: Boundary between the quantum and semi-classical regime, for a silica core fiber. The quantum regime stays above thefrontier.
C. Quantum switching.
To estimate the typical length needed for the soliton to tunnel from fiber to fiber, we reason as follows. We estimatefirst the time scale for the quantum tunneling. We split the wavefunction into the ‘right’ amplitude, Φ R , and theleft one, Φ L , each one being for the state in one fiber only. Because of the tunneling those states are not eigenstatesbut split into two eigenstates, one even (the ground state) Φ S and the other odd, Φ A , under permutation of the twofibers. One hasΦ L = (Φ S − Φ A ) / √
2, and Φ R = (Φ S + Φ A ) / √ A be half of this energy difference. If at time zero the soliton is on the right fiber, theevolution of its amplitude later on is given by2Φ( t ) = 1 √ S e − iE S t/ ~ + Φ A e − iE A t/ ~ ) . (35)Therefore the amplitude in one fiber oscillates with the period T osc ∼ h/ A. (36)In the following derivation, we approximate the energy splitting in each well by using the standard result for aparticle of momentum p ( x ) in a double-well:2 A = ~ ωπ exp ( − ~ Z a − a | p | dx ) (37)where [ − a, a ] is the x -range under the barrier for the given energy, and ω the pulsation of the wave-function in thebottom of the well. For a quadratic potential V ( x ), of curvature V ” around the minimum x M , the pulsation of aparticule of mass m is such that V ” = mω (38)The mass of the particle is deduced from its momentum under the barrier of height U , at x = 0, where p / m = U . Therefore the pulsation writes ω = s V ” U p (39)where all quantities are in physical units. Note that the dimensions are [ P ][ x ] = [ S ] and [ V ”] = [ V ] / [ x ], thereforethe dimension of x plays no role. For the fiber problem, we shall consider only one set of conjugate variables, ( x, Ψ),neglecting the η dependance of the potential, which plays a secondary role, moreover the physical quantities in equation(39) have to be expressed in terms of the reduced ones, and the ”time” period of equation (36) becomes a spatialperiod. This writes p = S (1) Ψ V ” = W (1) v ” = ω S (1) v ” U = W (1) ∆ VT osc = Zk ”0 /τ , where the curvature V ” and the height ∆ V of the potential barrier are deduced from equation (19), that gives∆ V = 0 .
037 + 6 κ , and v ” = κ − . Moreover the numerical results in Fig. 4 give Ψ = ln ( κ c /κ ). In the aboverelations Z is the ”true” spatial period along the optical axis of the fibers, obtained from equation (A1) after dividingall terms by k ”0 /τ to obtain a soliton of half-mass equal to unity as assumed in the present section.With these expressions, the equation (36) becomes ω T osc = Z ω k ”0 τ = 2 π Ψ √ v ”∆ V exp S/ ~ , (40)with S/ ~ = γ ( ω τ ) ln( κ c /κ ).Since the probability of finding the soliton in a given fiber oscillates with respect to the spatial variable z with awavelength Z , it also oscillates in time from one fiber to the other with the period τ = nZ/c , at a given z . Using thenumerical values given in the appendix for standard fibers, the period of the switching depends on the two parameters κ and τ . The frequency ν = c/nZ , and the spatial period Z are drawn in fig.(6), as function of the couplingparameter ratio κ c /κ . We have chosen two values of pulse duration, τ = 0 . ps (dashed line, corresponding to r = 1),and 1 . ps (solid line, r = 0 . κ c /κ is larger than few units where the WKBapproximation is valid, whereas the solid line corresponds to the ”purely quantum” regime. The two lines displayshigh frequencies, ranging from hundred of M hz , towards tens of
Ghz , that could be interesting for applications tohigh speed transmission. Note that while the solid line corresponds to the pure quantum regime, where the WKBapproximation used here is not valid, we infer that it could be possible that going beyond the WKB approximation,would lead to even higher frequencies. It could then lead to shorter switching lengths than those displayed in Fig.(6-b)). In the semi-classical regime, the switching length Z is longer, nevertheless it is much shorter than the halfperiod of switching in the CW linear case, L c = π κ phys . Indeed a pulse duration τ = 0 . ps , and a silica fiber, onehas L D = 16 m , that gives L c = 125 κ c κ when using k c = 0 . κ c κ = 100, the linear half period is L c = 12 . km , which is several order of magnitudes longer than the semiclassical switching length Z = 3 m (dashedcurve). (a)
20 40 60 80 100 κ c (cid:144) κ log H ν L (b)
20 40 60 80 100 κ c (cid:144) κ z FIG. 6: (a) Frequency of the periodic switching ν = 1 /τ , in Log scale, with ν in Hz ; (b) Spatial period Z , in m , as function ofthe ratio κ c /κ , for the case τ = 0 . ps (r=1, dashed line) and 1 . ps (r=0.1, solid line). For κ c κ = 100, the period is Z = 10 cm for the solid line, and about 3 m for the dashed line . V. SUMMARY AND DISCUSSION
Even though the tunneling phenomenon is very familiar in many wave-propagation problems, where the ”true”wave-vector −→ k becomes i −→ k after passing under a classical barrier, (as in the case of evanescent waves in the Fresneltheory), it appears in the present context in a slightly unusual form: starting from the classical model (6), (7) for thefield enveloppe, which looks strangely similar to the Schr¨odinger equation, our treatment based on the approximatetrial functions leads finally to the Euclidean system (26) which is not the Schr¨odinger equation for a wave function.Within the trial functions approximation, the WKB or quasiclassical limit gave us the possibility of estimatingrather easily the rate of quantum tunneling of a single soliton from one fiber to the other, even though it shouldremain classically in the same fiber forever. We found that this rate of tunneling is not small and could well be withinreach of present day-experiments.In the frame of the WKB approximation, we are trying to extend our results by getting rid of the trial functionsapproximation. Our aim is to check if the relation (29), that has been shown here to survive when going from two tofour unknown parameters in the trial functions approximation, is valid beyond this approximation. The calculationis heavier than the one presented here, because the time t is now considered as an infinite dimensional parameter,then the semi-classical trajectory must be calculated from a set of 4 coupled PDE’s, in place of the 4 coupled ODE’s(26) solved here. To derive these PDE’s, we can choose for example ( ℑ ( u ) , ℑ ( v )) and ( ℜ ( u ) , ℜ ( v )) as set of conjugatevariables ( p,q ) for the classical system (6), (7), with Hamiltonian H NLS = Z d t (cid:20) | ∂u∂t | − | u | + 12 | ∂v∂t | − | v | − κ ( uv + vu ) (cid:21) . (41)The Euclidean version of equations (6), (7) is then obtained by changing ( z, ℑ ( u ) , ℑ ( v )) into ( iz, i ℑ ( u ) , i ℑ ( v )). Theimportant point is that the tunneling factor does not depend on the choice of ( p,q ), while the Euclidean systemobviously does. Finally, the heteroclinic Euclidean trajectory is the solution connecting two of the classically permittedorbits, from z = −∞ to z = + ∞ . For a given energy E , these are defined by the integro-differential equation H NLS = E , where ℑ ( u ) = ℑ ( v ) = 0, and correspond to the two asymmetrical solitons.Because we have found that the Euclidean action S P hysE can be of order of ~ or even smaller in realistic experimentalconditions, it could even happen that the WKB quasiclassical approximation is not valid anymore for computing the4rate of transfer from one fiber to the other. Usually the order of magnitude of the action involved in the solitonpicture, even in a single fiber, is tacitly assumed to be far bigger than ~ , which is an assumption distinct from theone of a soliton made of many photons. Indeed the soliton picture addresses perturbations to this ‘bound state’ ofmany photons that may be small enough to imply variations of the action of order of ~ , and so require some sortof (‘second’) quantization. We plan to come to this general question in future work, and outline here some of theestimated problems.A treatment using the trial function, but valid beyond the WKB approximation, is obviously more complicatedthan what we did here, and perhaps questionable. Indeed it needs to consider both the trial functions and theirparameters as operators. Moreover it amounts to assume that the fluctuations in t and z are decoupled, and, lastbut not least, our result derived in the WKB approximation likely signals that the assumption behind the classical(meaning non quantum) theory for describing soliton in coupled fibers does not hold anymore and that the quantumpicture has to be used from the start, which makes it theoretically challenging.We assumed that every phenomenon under study involved solitons seen as a coherent quantum objects. We arguedthat this requires that any typical time, the tunneling time in particular, is far shorter than the coherence time.This coherence time is of order of t c /N , with t c coherence time of a single photon in the soliton, i.e. its mean-freeflight time without change in phase or frequency. Because of the division by N this may be a very short time. Attimes longer than the coherence time any physical effect related to the quantum coherence between states of solitonspropagating in either fiber is washed out. The final state, as described in the density matrix formalism, is a stateof equal probability of the soliton on either side without nondiagonal element. The experimental manifestation ofthis state will be a probability 1 / κ periodically modulated in z , in order to stimulate the switching process. Acknowledgments
Elisabeth Ressayre et Jean Ginibre are gratefully acknowledged for stimulating discussions, and Laurent Di Menzafor providing us a code for simulations of the NLS model with transparent boundary conditions.
APPENDIX A: PHYSICAL UNITS
This appendix is about the relationship between quantities measured in physical units for a standard fiber carryingsolitons and the dimensionless quantities used in the bulk of our paper. We relate first the number of photons in atypical soliton to its time duration, a duration called τ that we shall use afterwards to give various order of magnitudespertinent to our problem.We use the standard expression of the electric field in a fiber, written with the same notations as in the book byNewell and Moloney [15]. The electric field of the EM wave in the fiber is modelized by the wave-packet expression, E = R ( x t ) B ( z, t ) exp( iω t − k z ) + c.c. which obeys in a first approximation Maxwell’s equations, when the durationof the pulse is not too short, x t being the transverse coordinate, and R ( x t ) the dimensionless radial amplitude with R | R ( x ′ ) | dx ′ = σ as the core area.Taking δωω as a small parameter, δω being the frequency width of the pulse, one obtains the NLS equation writtenin variables z, τ = t − z/v g , the nonlinear and the dispersion term having opposite signs: i ∂B∂z + k ”0 ∂ B∂τ + 2 πn ′ λ | B | B = 0 . (A1)5Note that we turned to the standard writing of the coefficients of the NLS equation, n ′ being the modulus of thecoefficient of the cubic Kerr effect and k ”0 the modulus of the second derivative of the wavenumber k with respect tothe the frequency of the EM wave.The soliton solution is B ( z, t − z/v g ) = B m sech (( t − z/v g ) /τ ) , (A2)with β | B m | = 1 /τ . (A3)where β = πn ′ λ k ”0 . Its energy is W = Z P ( t ) dt = σ Z I ( t )d t, (A4)where I ( t ) is the optical intensity measured in watt per square meter, and P ( t ) is the Poynting vector integratedacross the fiber section, with a result expressed in Watts: P ( t ) = Z ( E ∧ H ) . z dS (A5)where z is the unit vector in the direction of propagation.The magnetic field in the wave (supposing it is linearly polarized with the electric field in the x -direction) is H x = ncε E y , n index of refraction. For a material with instantaneous response, after averaging over one period ofthe field oscillations, one finds P ( t ) = 2 cε σn | B ( t ) | . (A6)At leading order, i.e. by taking into account the linear part of refractive index, n = n , this yields I (0) ( t ) = 2 cε n | B ( t ) | . (A7)Whence the energy of the pulse is: W (0) = N ~ ω = 2 n cε σ Z | B ( t − z/v g ) | dt = 4 n cε σ λ k ”0 πn ′ τ . (A8)Finally the relationship between the photon number and the pulse duration writes N τ = 4 n ε σ ( λ π ) k ”0 n ′ ~ . (A9)Similarly the action is S (0) = N ~ = W (0) /ω, (A10)at leading order.In MKSA units, with standard values (see [15]) of optical fibers composed of silica cores, this gives: n = 1 . ǫ = 0 . . − F/m , or ǫ c = 1 /Z with Z = 377Ω the impedance of free space, ~ = 10 − J.sλ = 1 . . − mσ ∼ − for a 10 µm -area fiber. k ”0 = 2 . . − s /mn ′ = 2 . . − ( m/V ) With these data, the number of photons in the pulse of duration τ , measured in seconds obeys the relation:6 N τ ∼ . − , (A11)that gives N ∼ . photons for a ps -pulse.Let us note that the nonlinear index of refraction n ′ may be several orders of magnitude larger, when using othermaterials. For example, in the experiment of Wa et al. [18], the optical switch was studied in multiple quantum wellwave-guides, with n ′ = 10 − ( m/V ) . Coherent part of the energy and action
Let us write the energy and action as W = W (0) + W (1) , (A12)and S = S (0) + S (1) . (A13)The dominant contributions are proportional to the linear part of the refractive index, namely a term containedin the Maxwell equation. The subdominant contributions W (1) and S (1) , correspond to the terms contained in theenvelope equation, they are perturbations to the dominant effects calculated above. For two coupled fibers, theseperturbations result from balanced effects of dispersion, coupling and nonlinearity. They are proportional to theenergy and action of the the dimensionless NLS equation (6-7), ( W (1) = w (1) H NLS S (1) = w (1) ω S NLS (A14)where the scaled energy H NLS is defined in equation (41) , and the action in equation (10).The coefficient w (1) may be calculated by using the expression of the Poynting vector (A6) valid for dispersionlessKerr media, where n = n + n I = n + n ′ | B ( t ) | , (A15)This gives W (1 ,Kerr ) = 2 n ′ cǫ σ R dt | B ( t ) | .Taking the hyperbolic secant solution (A2-A3), one obtains W (1 ,Kerr ) = 2 n ′ cǫ σ | B M ( t ) | τ R dtsech ( t ), or W (1 ,Kerr ) = − n ′ cǫ σ β τ H KerrNLS , (A16)where H KerrNLS is the Kerr contribution of the Hamiltonian (second term in the r.h.s. of equation 5). This correction isthe ”coherent” part of the energy in the sense that it is proportional to the square of the intensity, or of the photonnumber. Finally we are ready to express the physical value of the Hamiltonian and action associated to a solitonwhose temporal width is scaled to τ as it was assumed in sections 3-4, by using the expression (A14) with w (1) ≃ n ′ cǫ σ β τ . (A17)This allows in particular to express concretely the constraint that nonlinear effects are small, that is that W (1) < M´ecanique Quantique,Th´eorie non relativiste , (Ed. Mir, Moscou ,1966).[6] B. A. Malomed, I. M. Skinner, P. L. Chu and G. D. Peng, , Phys. Rev. E , 4084 (1996); I. M. Uzunov, R. Mushall, M.Golles, Yu. S. Kivshar, B. A. Malomed, and F. Lederer, Phys. Rev. E , 2527 (1995).[7] S. Trillo, S. Wabnitz, E. M. Wright and G.I. Stegeman, Opt. Comm. , 166 (1989); E. M. Wright, G.I. Stegeman and S.Wabnitz, Phys. Rev. A , 4455 (1989); S. Trillo, S. Wabnitz, E. M. Wright and G.I. Stegeman, Opt. Lett. A , 871(1988); and Opt. Lett. A , 672 (1988)[8] C. Par´e, M. Florjanczyk, Phys.Rev. A , 6287 (1990).[9] Y.S. Kivshar, Opt. Lett. , 7 (1993).[10] N. F. Smyth, A. L. Worthy, J. Opt. Soc. Am. , 2610 (1997); N. F. Smyth , A. H. Pincombe, Phys. Rev. E , 7231(1998).[11] Y. Pomeau, Europhys. Lett., 951 (2006).[12] Y. Pomeau, Nonlinearity , 707 (1992).[13] L. D. Landau and E. M. Lifchitz in Mechanics and Electrodynamics (Pergamon Press, N.Y., 1972).[14] S.Coleman, in Proc. Int. School of Subnuclear Physics , Europhys. (Erice , 2006); and in Aspects of symmetry (CambridgeUniversity Press, 1985).[15] A. C. Newell and J.V. Moloney Nonlinear Optics , ( Addison-Wesley Publ. Company, 1992).[16] A. Sizmann, Appl. Phys. B , 745 (1997).[17] E. J. Post, Rev. of Mod. Phys. , 475 (1967).[18] P. Li Kam Wa, J.E. Sitch, N. J. Mason, J. S. Roberts, P. N. Robson, Electronics Letters,21