Multi-mode solitons in the classical Dicke-Jaynes-Cummings-Gaudin Model.

# Multi-mode solitons in the classical Dicke-Jaynes-Cummings-Gaudin Model.

Abstract: We present a detailed analysis of the classical Dicke-Jaynes-Cummings-Gaudin integrable model, which describes a system of spins coupled to a single harmonic oscillator. We focus on the singularities of the vector-valued moment map whose components are the mutually commuting conserved Hamiltonians. The level sets of the moment map corresponding to singular values may be viewed as degenerate and often singular Arnold-Liouville torii. A particularly interesting example of singularity corresponds to unstable equilibrium points where the rank of the moment map is zero, or singular lines where the rank is one. The corresponding level sets can be described as a reunion of smooth strata of various dimensions. Using the Lax representation, the associated spectral curve and the separated variables, we show how to construct explicitely these level sets. A main difficulty in this task is to select, among possible complex solutions, the physically admissible family for which all the spin components are real. We obtain explicit solutions to this problem in the rank zero and one cases. Remarkably this corresponds exactly to solutions obtained previously by Yuzbashyan and whose geometrical meaning is therefore revealed. These solutions can be described as multi-mode solitons which can live on strata of arbitrary large dimension. In these solitons, the energy initially stored in some excited spins (or atoms) is transferred at finite times to the oscillator mode (photon) and eventually comes back into the spin subsystem. But their multi-mode character is reflected by a large diversity in their shape, which is controlled by the choice of the initial condition on the stratum.

## 1 Introduction.

The Dicke model has been used for more than fifty years in atomic physics to describe the interaction of an ensemble of two-level atoms with the quantized electromagnetic field [1]. Recently, many experimental and theoretical works have considered systems in an optical cavity where matter is coupled to a single eigenmode of the field [2, 3]. These systems offer some unprecedented opportunities to realize quantum bits and to process quantum information. The integrable version of the monomode Dicke model considered in this paper [4, 5] is obtained by making the so called rotating wave approximation in which the non resonant terms (e.g. for which a photon is created while an atom is excited) are discarded. In the case of a single atom, this model has been solved by Jaynes-Cummings [6], and has been used to study the effect of field quantization on Rabi oscillations [7]. In the many atom case, besides the development of cavity QED [2] and circuit QED [3] already mentioned, a surge of interest in the Dicke-Jaynes-Cummings-Gaudin model has been motivated in the context of cold atoms systems, where a sudden change in the interactions between atoms is achieved by sweeping the external magnetic field through a Feshbach resonance [8, 9]. The subsequent dynamics of the atomic system is well described by the DJCG model prepared in the quantum counterpart of a classically unstable equilibrium state.

This motivates the study of unstable equilibrium points in classical and quantum integrable models. From the classical viewpoint, an integrable system with degrees of freedom is characterized by a collection of mutually commuting functions ,,…, over the -dimensional phase-space. Usually, the physical Hamiltonian is expressed as a function of these conserved quantities. Besides the classical trajectories generated by , it is of great interest to consider the so-called moment map, which associates to any point in phase space the -vector . The image of the moment map is of great importance. Fig.[1] provides an example of such an image of the moment map for the two spins Dicke-Jaynes-Cummings-Gaudin model. To each point in this image corresponds a level set in phase space, that is to say the set of points such that . For many systems of physical relevance these level sets are compact (as it is the case here). In the case of a regular value of the moment map, the Arnold-Liouville theorem states that the corresponding level set is an -dimensional torus. Non regular values of the moment map, (for which the rank of the moment map may be ), are in some sense more interesting, because they correspond to special torii whose dimension may be lower than or which may contain singular submanifolds. The image of the moment map and its critical values encode all the information on the fibration of phase space in Liouville torii. It is important to realize that the level set of a non-regular value of the moment map is in general a stratified manifold on which the rank of the moment map may jump. The dimension of each stratum is equal to the rank of the differential of the moment map on that stratum.

An extreme case is given by the level set containing an equilibrium point. By definition, the gradient of the moment map vanishes at such point (the rank is zero), which is then an equilibrium point for any Hamiltonian expressed in terms of , ,…,. For a purely elliptic equilibrium point, the level set is reduced to this point. But for an unstable point, the level set contains the equilibrium point, of course, but also a whole manifold whose description is far from trivial. In the vicinity of the unstable critical point, the stratification can be qualitatively understood by the quadratic normal form [10], deduced from the Taylor expansion of the moment map to second order in the deviations from the critical point. A detailed derivation of these normal forms in the Dicke-Jaynes-Cummings-Gaudin model has been presented in our previous contribution [11]. But the information contained in the normal form is purely local, and it is not sufficient to describe globally the level set containing an unstable critical point. Global characteristics can be access by viewing these level sets as collections of trajectories, solutions of the equations of motion. In this paper, we shall present explicit solutions of the physical Hamiltonian equations of motion for an arbitrary initial condition chosen on such a critical level set. They can be described as solitonic pulses in which the energy initially stored in some excited spins (or atoms) is transferred at finite times to the oscillator mode (photon) and eventually comes back into the spin subsystem.

We achieve this by using the well known algebro-geometric solution [13, 15, 14] of the classical Dicke-Jaynes-Cummings-Gaudin model. A key feature of this approach is that the knowledge of the conserved quantities ,,…,, (equivalently a point in the image of the moment map) is encoded in a Riemann surface, called the spectral curve. It is of genus , where is the number of spins. The state of the system can be represented by a collection of points on the spectral curve. Remarkably, the image of this collection of points by the Abel map follows a straight line with a constant velocity on the Jacobian torus associated to the spectral curve. This construction linearizes the Hamiltonian flows of the model.

It is well known that critical values of the moment map correspond to degenerate spectral curves [16, 13], and this seems to be the most efficient way to locate the critical values of the moment map. This strategy has been used extensively in our previous discussion of the moment map in the classical Dicke-Jaynes-Cummings-Gaudin model [11]. We believe that the separated variables, which we will use in their Sklyanin presentation [15], are a powerful tool to uncover the stratification of a critical level set. In this paper, we show that each stratum of dimension on such level set is obtained by freezing separated variables on some zeros of the polynomial which defines the spectral curve.

We will aslo pay a special attention to the problem of finding the physical slice on each stratum. This is not a straightforward question, because the algebro-geometric method is naturally suited to solve equations of motion in the complexified phase-space. But only solutions for which all spin components and the two oscillator coordinates are real are admissible physically. The determination of the physical slice for a generic regular value of the moment map remains an open problem. But on the critical level sets of the moment map, corresponding to a spectral curve of genus zero, the problem greatly simplifies. This corresponds to level sets of the moment map containing critical points of rank zero or one for which we are able to derive an explicit solution for an arbitrary number of spins. Remarkably these solutions were already constructed by Yuzbashyan [12], and they find here their place in a broader picture aiming at organizing all particular solutions of the Dicke-Jaynes-Cummings-Gaudin model.

This paper is organized as follows. Section 2 presents the classical Dicke-Jaynes-Cummings-Gaudin model of spins coupled to a single harmonic oscilator, together with the main features of the algebro-geometric solution. In particular, we emphasize that the separated variables capture in a very natural way the stratification of a critical level set, each stratum being characterized by a subset of frozen variables on some double roots of the spectral polynomial. These notions are then illustrated in section 3 on the example of the system with only one spin. In this case, the reality conditions can be worked out explicitely, both on the critical level sets associated to unstable equilibrium points and for generic values of the moment map. Section 4 gives the complete determination of the real slice for the level set of an equilibrium point, and for arbitrary values of . This corresponds to the case where all the roots of the spectral polynomial are doubly degenerate. We recover here the normal solitons first constructed by Yuzbashyan [12]. Section 5 addresses the same question, in the slightly more common situation where the spectral polynomial has doubly degenerate roots and two simple roots. In this case, we are describing a level set whose smallest stratum has dimension 1. The corresponding values of the moment map lie on curves in its -dimensional target space. We benefit in this case from the fact that the spectral curve remains rational. We get therefore a general formula for the anomalous solitons in Yuzbashyan’s terminology. Section 6 illustrates this general theory for the system with two spins, and section 7 for the system with three spins. We present explicit examples of solitonic pulses corresponding to solutions living in strata of dimensions 2, 3, and 4. The presence of several directions of unstability away from the corresponding equilibrium points is reflected by the non-monotonous time dependence seen in these pulses, which contrasts with the simple shape observed for the model with a single spin. Our conclusions are stated in section 8. Finally, an Appendix provides a brief summary of the construction [11] of quadratic normal forms of the moment map in the vicinity of equilibrium points.

## 2 The classical Dicke-Jaynes-Cummings-Gaudin model.

### 2.1 Integrability

This model, describes a collection of spins coupled to a single harmonic oscillator. It derives from the Hamiltonian:

 H=n∑j=1(2ϵj+ω)szj+ω¯bb+n∑j=1(¯bs−j+bs+j) (1)

The are spin variables, and is a harmonic oscillator. The Poisson brackets read:

 {saj,sbj}=−ϵabcscj,{b,¯b}=i (2)

The brackets are degenerate. We fix the value of the Casimir functions

 →sj⋅→sj=s2

Phase space has dimension . In the Hamiltonian we have used which have Poisson brackets . The equations of motion read:

 ˙b = −iωb−in∑j=1s−j (3) ˙szj = i(¯bs−j−bs+j) (4) ˙s+j = i(2ϵj+ω)s+j−2i¯bszj (5) ˙s−j = −i(2ϵj+ω)s−j+2ibszj (6)

Integrability is revealed after introducing the Lax matrices:

 L(λ) = 2λσz+2(bσ++¯bσ−)+n∑j=1→sj⋅→σλ−ϵj (7) M(λ) = −iλσz−iω2σz−i(bσ++¯bσ−) (8)

where are the Pauli matrices, .

 σx=(0110),σy=(0−ii0),σz=(100−1),

It is not difficult to check that the equations of motion are equivalent to the Lax equation:

 ˙L(λ)=[M(λ),L(λ)] (9)

Letting

 L(λ)=(A(λ)B(λ)C(λ)−A(λ))

we have:

 A(λ) = 2λ+n∑j=1szjλ−ϵj (10) B(λ) = 2b+n∑j=1s−jλ−ϵj (11) C(λ) = 2¯b+n∑j=1s+jλ−ϵj (12)

These generating functions have the simple Poisson brackets:

 {L1(λ),L2(μ)}=−i[P12λ−μ,L1(λ)+L2(μ)]

where

 P12=⎛⎜ ⎜ ⎜⎝1000001001000001⎞⎟ ⎟ ⎟⎠

It follows immediately that Poisson commute for different values of the spectral parameter:

 {Tr(L2(λ1)),Tr(L2(λ2))}=0

Hence

 Λ(λ)≡12Tr(L2(λ))=A2(λ)+B(λ)C(λ)

generates Poisson commuting quantities. One has

 Λ(λ)=Q2n+2(λ)∏j(λ−ϵj)2=4λ2+4Hn+1+2n∑j=1Hjλ−ϵj+n∑j=1s2(λ−ϵj)2 (13)

where is a polynomial of degree . The commuting Hamiltonians , read:

 Hn+1=b¯b+n∑j=1szj (14)

and

 Hj=2ϵjszj+(bs+j+¯bs−j)+∑k≠jsj⋅skϵj−ϵk,j=1,⋯,n (15)

The physically interesting Hamiltonian eq.(1) is:

 H=ωHn+1+n∑j=1Hj (16)

On the physical phase space the complex conjugation acts as and , and of course is the complex conjugate of . Hence for real, is real and . It follows that on the physical phase space, is positive when is real.

### 2.2 Separated variables.

The Lax form eq. (9) of the equation of motion implies that the so-called spectral curve , defined by is a constant of motion. Specifically:

 Γ:μ2−A2(λ)−B(λ)C(λ)=0, i.e. μ2=Q2n+2(λ)∏j(λ−ϵj)2 (17)

Defining , the equation of the curve becomes which is an hyperelliptic curve. Since the polynomial has degree , the genus of the curve in . Because the model is integrable, it is possible to construct action-angle coordinates (at least locally), but their connection to the initial physical dynamical variables is rather complicated. In this work, we prefered to work with the so-called separated variables which have the double advantage that their equations of motion are much simpler than the original ones and that they are not too far from the physical spin and oscillator coordinates. The separated variables are points on the curve whose coordinates can be taken as coordinates on phase space. They are defined as follows. Let us write

 C(λ)=2¯b+n∑j=1s+jλ−ϵj≡2¯b∏nk=1(λ−λk)∏nj=1(λ−ϵj) (18)

the separated coordinates are the collection . They have canonical Poisson brackets

 {λk,μk′}=−iδk,k′

Notice however that if theses variables are not well defined.

There are only such coordinates which turn out to be invariant under the global rotation generated by :

 b→eiθb,¯b→e−iθ¯b,s−j→eiθs−j,s+j→e−iθs+j

So they describe the reduced model obtained by fixing the value of and taking into consideration only the dynamical variables invariant under this action. The initial dynamical model can be recovered by adding the phase of the oscillator coordinates to the separated variables. The equations of motion with respect to in these new variables are (no summation over ):

 ∂tiλk=iμk(B−1)ik,Bkj=1λk−ϵj,(B−1)jp=∏l≠p(ϵj−λl)∏i(λp−ϵi)∏i≠j(ϵj−ϵi)∏l≠p(λp−λl) (19)

Using the identities:

 n∑i=1ϵpi∏j≠i(ϵi−ϵj)=δp,n−1,0≤p≤n−1

the flow associated to the physical Hamiltonian eq.(16) reads:

 ∂tλk=iμk∏j(λk−ϵj)∏l≠k(λk−λl),μk=A(λk) (20)

The equations of motion eq.(20) must be complemented by the equation of motion for which allows to recover the motion of the full model from the motion of the separated variables in the reduced model. We have:

 ∂t¯b=iω¯b+i∑Res ϵjC(λ)=iω¯b−i∑Res ∞C(λ)=i¯b(ω−2(Σ1−σ1(ϵ))) (21)

where and . Of course, we also have the complex conjugated equation of motion for .

We show now how to reconstruct spin coordinates from the separated variables and the oscillator coordinates . For we have eq.(18). It is a rational fraction of which has simple poles at whose residue is . For , we write:

 A(λ)=Pn+1(λ)∏nj=1(λ−ϵj)

The polynomial is of degree , and we know its value at the points because:

 Pn+1(λj)=μjn∏k=1(λj−ϵk) (22)

Therefore, we can write:

 Pn+1(λ)=2λn∏j=1(λ−ϵj)+∑i(μi−2λi)n∏k=1(λi−ϵk)∏l≠i(λ−λl)∏l≠i(λi−λl) (23)

Once and are known, we can find the spin themselves by taking the residues at . We get:

 s+j=2¯b∏ni=1(ϵj−λi)∏nk≠j(ϵj−ϵk),szj=∏nl=1(ϵj−λl)∏nk≠j(ϵj−ϵk)∑i[(2λi−μi)∏k≠j(λi−ϵk)∏l≠i(λi−λl)] (24)

The modulus of is invariant under the global action generated by and is obtained from (recall that in the reduced model is a parameter)

 ¯bb=Hn+1−∑jszj

The phase of and is not determined in the reduced model. The same phase appears in the formula for . Finally, the components are obtained from the constraints:

 (szi)2+s+is−i=s2

They are determined up to the phase of .

Note that if the ’s are arbitrary complex numbers, is not the complex conjugate of , as it is the case for the physical phase-space. This shows that separated variables are natural coordinates on the complexified phase-space. In these variables, the problem of identifying the sets corresponding to physical configurations (i.e. those for which all spin components are real) is rather non-trivial, and most of the present work is dedicated to it.

In many situations, and in particular in this article, we are interested in the system with prescribed real values of the conserved quantities, ,…,, i.e. we take as coordinates the instead of the . This amounts to fixing the Liouville torus we work with, or equivalently the spectral polynomial . In this setting, the are determined by the equation

 n∏k=1(λj−ϵk)μj=±√Q2n+2(λj)

The choice of the signs in this formula is very important and will be a recurrent theme in our subsequent discussions. They affect the formulae for or equivalently the polynomial . The values of ’s can then be obtained from which is determined by:

 B(λ)=Q2n+2(λ)−P2n+1(λ)C(λ)∏k(λ−ϵk)2=12¯bQ2n+2(λ)−P2n+1(λ)∏i(λ−λi)∏k(λ−ϵk) (25)

The polynomial in the numerator is of degree , and moreover it is divisible by because , so we can write

 B(λ)=2b∏i(λ−¯λi)∏k(λ−ϵk) (26)

The spin components are obtained by taking the residue at :

 s−j=2b∏i(ϵj−¯λi)∏k≠j(ϵj−ϵk)

In this consruction, the variables and the corresponding are complicated functions of the and of the choice of signs used in the determination of the variables . The set of physical configurations (often refered to here as the real slice) is obtained by writing that the set is the complex conjugate of the set . This leads to a set of complicated relations whose solution is not known in general. It is the purpose of this work to show how to implement them in particular cases.

### 2.3 Moment map and degenerate spectral curves

In this section, we discuss the case of a spectral polynomial having double roots, so that can be written as:

 Q2n+2(λ)=4p2m+2(λ)n−m∏i=1(λ−Ei)2,p2m+2(λ)=λ2m+2+b2m+1λ2m+1+⋯ (27)

The are either real or come in complex conjugated pairs. Moreover is positive for real . As discussed in some previous works [5, 11, 16], this corresponds to critical values of the moment map. The preimage of the moment map for such value is a stratified manifold, whose various strata have dimensions ranging from to . Separated variables provide a very natural access to this stratification, because each stratum of dimension can be obtained by freezing separated variables on double zeroes of . On such stratum, the rank of the moment map also drops to . As explained before [5], the dynamics of the system for initial conditions lying in this stratum is very similar to the one of an effective model with spins, but we shall not use this physically appealing result here.

To produce a polynomial of the form eq.(27) is not completely straightforward because it must also be compatible with eq.(13). Let us explain how it works.

The fact that the rational fraction has double poles at whose weight is imposes conditions:

 ∏i(λ−Ei)|ϵj=sαj2√p2m+2(ϵj)∏k≠j(ϵj−ϵk)

where . The values of the polynomial at the points are therefore known, and we can write by Lagrange interpolation formula:

 ∏i(λ−Ei)=∑jsαj2√p2m+2(ϵj)∏i≠j(λ−ϵi) (28)

But the degree of the left hand side is while the degree on the right hand side is superficially , so we have the consistency conditions:

 ∑jαjϵkj2√p2m+2(ϵj)=0,k=0,⋯,m−2 (29)

and:

 ∑jsαjϵm−1j2√p2m+2(ϵj)=1 (30)

These equations are obtained by writing that:

 ∮dz2πizk∏n−mi=1(z−Ei)∏ni=1(z−ϵi)=δk,m−1,k=0,⋯,m−1

where the integrals are taken along a contour in the complex plane which encircles all the ’s. These are exactly conditions on the coefficients of , of which the leading coefficient is known. One more condition is obtained by writing that in the right-hand side of eq. (13) the coefficient of vanishes. This gives:

 b2m+1+2(σ1(ϵ)−σ1(E))=0

But we also have:

 ∮dz2πizm∏n−mi=1(z−Ei)∏ni=1(z−ϵi)=σ1(ϵ)−σ1(E)=∑jsαjϵmj2√p2m+2(ϵj)

where the integral is again on a contour encircling all the ’s. Assembling the previous two equations gives the consistency condition:

 b2m+1+∑jsαjϵmj√p2m+2(ϵj)=0 (31)

Altogether, we are left with free coefficients in which play the role of conserved Hamiltonians of an effective system with spins [5]. All the other symmetric functions , are then determined by eq.(28).

If , that is to say , eq.(28) must be slightly modified as

 ∏i(λ−Ei)=(λ+β)∏j(λ−ϵj)+∑jsαj2∏i≠j(λ−ϵi) (32)

This means

 Q2n+2(λ)∏j(λ−ϵj)2=(2λ+2β+∑jαjsλ−ϵj)2 (33)

Comparing with eq.(13) at we see that .

Let us discuss now the freezing of the at the double roots of . The equations of motion eq.(20) become

 ∂tλk=±i∏j(λk−Ej)√p2m+2(λk)∏l≠k(λk−λl), (34)

so if at , it stays there forever, and it is consistent with the equations of motion to freeze some at the double roots of . This however cannot be done in an arbitrary way as we now explain.

Let us assume that has a real root at . This means that vanishes when . But for real , is real and one has so that , , must all vanish at . In particular, recalling eq.(18), this means that one of the separated variables, say , is frozen at the value , provided . This implies also that divides simultaneously , and , and therefore divides . A single real root of is necessarily a double root. Writing: , , , , we see that:

 ˜Q2n(λ)=∏j(λ−ϵj)2(˜A(λ)2+˜B(λ)˜C(λ))

so we can apply the same discussion to the real roots of . We deduce from this that the multiplicity of the real root of is even and that divides simultaneously , and . In this case, separated variables ,…, are frozen at the real value .

Nothing as simple holds in the case of a simple complex root. So, let us assume that there is a double complex root . Of course the complex conjugate is also a double root. What can be said in general is that the values of ,…, freeze by complex conjugated pairs.

In fact we have hence

 Λ′(λ)=2A(λ)A′(λ)+B′(λ)C(λ)+B(λ)C′(λ)

On a degenerate spectral curve, has at least one double zero . Therefore

 0 = A2(E)+B(E)C(E) 0 = 2A(E)A′(E)+B′(E)C(E)+B(E)C′(E)

By contrast to the real case, we cannot infer from these equations that . This is related to the fact that for critical spectral curves associated to unstable critical points, the corresponding level set of the conserved Hamiltonians has dimension equal to the number of unstable modes. So we expect that some separated variables can remain unfrozen. But if one of them freezes at , that is if , the first equation implies and the second equation implies . Therefore if the zero of at is simple, then necessarily . But is the complex conjugate of which must therefore vanish provided . From this we conclude that another separated variable freezes at .

In contrast to the real case, freezing is not compulsory, but it is the possibility to freeze the by complex conjugated pairs that leads to the description of the real slice and the stratification of the level set.

## 3 The one-spin model.

The model with a single spin coupled to the oscillator is interesting, because it illustrates most of the points discussed so far. In this case, there are only two conserved quantities, and , which read:

 H1 = 2ϵ1sz1+bs+1+¯bs−1 (35) H2 = ¯bb+sz1 (36)

The rank of the momentum map can be either 0, 1, or 2. The later case corresponds to generic Arnold-Liouville tori of dimension 2. Let us now recall briefly the discussion of critical values of the moment map given in a previous work [11].

### 3.1 Rank zero

The rank of the moment map vanishes on the critical points given by . Hence we have two points:

 sz1=e1s,e1=±1

The corresponding values are:

 P1(↑)=[2ϵ1s,s],P2(↓)=[−2ϵ1s,−s]

To determine the type of the singularities, we look at the classical Bethe equations (see section 9) which read:

 2E+se1E−ϵ1=0⇔2E2−2ϵ1E+se1=0

The discriminant of this equation is . When the spin is down (point ), the discriminant is positive, the two classical Bethe roots are real and this is an elliptic singularity, in agreement with the general analysis of section 9. When the spin is up (point ) we have real roots when (i.e. the singularity is elliptic in this case), and a pair of complex conjugate roots when (i.e. the singularity is focus-focus in that case). The image of the moment map is shown on Fig.[2] in the unstable case. We see that the stable critical point is located at the tip of the image of the moment map, whereas the unstable one lies in the interior of this domain.

In the one spin case, eq.(13) reads :

 Q4(λ)(λ−ϵ1)2=4λ2+4H2+2H1λ−ϵ1+s2(λ−ϵ1)2 (37)

For the values it is completely degenerate and the spectral polynomial has two double roots , identical to the classical Bethe roots, and reads:

 Q4(λ)=4(λ2−ϵ1λ+s2e1)2=4(λ−E1)2(λ−E2)2

where

 E1+E2=ϵ1,E1E2=se12

Let us now describe in more detail the real slices corresponding to the two critical values of . Recall that

 sz1=(ϵ1−λ1)(2λ1−μ1) (38)

We express first the separated variable as a function of :

 μ1=±2(λ1−E1)(λ1−E2)(λ1−ϵ1) (39)

We have to distinguish the stable and unstable cases. In the stable case and are real and we know that has to be frozen at one of them, say . Then , there is no problem of sign in eq.(39) and

 sz1=2E1(ϵ1−E1)=2E1E2=se1

as it should be. The level set in this case is composed of only one stratum consisting of the critical point itself where the rank of the moment map is zero.

Let us now assume that we are in the unstable case, that is at the point , and . In this case are complex and remains unfrozen. Let us choose the sign in eq.(39). Inserting into eq.(38) we obtain

 sz1=se1

so that, in the spin variables, we are precisely at the critical point. On this stratum of the level set the rank of the moment map is zero. It is very interesting to remark that the variable completely disappears in this case (hence the rank zero). In terms of the separated variable, this stratum of the level set seems therefore to consists of the whole -plane. This is due to the fact that at the critical point and the separated variables are not well defined and appear as singular coordinates. However, as we will show below, the coordinate has the magic property of realizing a blowup of the singularity.

 sz1=−se1−4λ1(λ1−ϵ1)

Writing that is real, , gives

 λ1+¯λ1=ϵ1 (40)

Setting , we see that the real slice is given by the vertical line located at . It is not the whole line however. We find from

 ¯bb=2s−ϵ21−4y2≥0

The real slice of the reduced model corresponds in this case to the segment parallel to the imaginary axis delimited by the two double roots and of the spectral polynomial. This manifold has real dimension one.

As we have discussed before [11, 18], this corresponds to a real slice of the model reduced by the global action eq.(21). From the viewpoint of the original model, we have to reintroduce this global angle and the real slice becomes a two dimensional torus pinched at the unstable critical point.

Hence in this simple case we see that the level set of the unstable critical point is composed of two strata : the critical point itself where the rank of the moment map is zero and the pinched torus where the rank is two.

The dynamics of the model on this torus is the composition of a large motion in which the oscillator amplitude goes to zero at time but reaches a finite maximum at a finite time, and a global rotation. Only the former movement is captured by the reduced model, and it maps into the finite segment of the variable which we have just described. To see this, let us consider the equation of motion for the flow generated by :

 ∂t1λ1=2i(λ1−E)(λ1−¯E) (41)

whose solution is:

 λ1=E−¯EX1−X,X=X0e2i(E−¯E)t1 (42)

The reality condition eq.(40) becomes:

 (E−¯E)(X−¯X)=0

which imposes so that is real. Its absolute value can be absorbed in the origin of time . Only its sign matters. The constraint is equivalent to , and we recover the fact that runs along the line interval joining and .

### 3.2 Rank one

Physical configurations at which the rank of the moment map is equal to one correspond to a spectral polynomial with one double zero, which is necessarily real. We therefore write:

 Q4(λ)=(λ+a0)2(b2λ2+b1λ+b0) (43)

where and are real. We denote:

 a0=12x−ϵ1

Taking into account the usual constraints on saying that it depends only on two free parameters and , see eq.( 37), we can determine , , and in terms of by solving linear equations. We find:

 b2=4,b1=−4x,b0=4ϵ1x3−ϵ21x2+s2x2
 H1 = −x4−2ϵ1x3−4s22x H2 = −3x4−8ϵ1x3+4ϵ21x2−4s24x2

These are parametric equations for a curve which coincides with the boundaries of the image of the moment map in the plane, for which an illustration can be found on Fig.[2].

The determination of the real slice is straightforward in this case. Indeed, because we have a real double root, the separated variable is frozen at the value . This single point of the reduced model corresponds to the one-dimensional orbits under the global rotations generated by characterized by a common phase on and :

 sz1 = −12x(x−2ϵ1)