We derive and study the hyperbolic reduced Maxwell-Bloch equations (HRMB) which acts as a simplified model for the dynamics of strongly correlated Bose-Einstein condensates. A proof of their integrability is found by the derivation of a Lax pair which is valid for both the hyperbolic and standard cases of the reduced Maxwell-Bloch equations. The origin of the latter lies in quantum optics. We derive explicit solutions of the HRMB equations that correspond to kinks propagating on the Bose-Einstein condensate (BEC). These solutions are different from Gross-Pitaevskii solitons because the nonlinearity of the HRMB equations arises from the interaction of the BEC and excited atoms

Arnaudon, A

Gibbon, JD

English

Spiral - Imperial College Digital Repository

arXiv:1609.00690v1  [math-ph]  2 Sep 2016Integrability of the hyperbolic reduced Maxwell-Bloch equations for stronglycorrelated Bose-Einstein condensatesAlexis Arnaudon∗ and John D. GibbonDepartment of Mathematics, Imperial College, London SW7 2AZ, UK.(Dated: September 5, 2016)We derive and study the hyperbolic reduced Maxwell-Bloch equations (HRMB) which acts as asimplified model for the dynamics of strongly correlated Bose-Einstein condensates. A proof of theirintegrability is found by the derivation of a Lax pair which is valid for both the hyperbolic andstandard cases of the reduced Maxwell-Bloch equations. The origin of the latter lies in quantumoptics. We derive explicit solutions of the HRMB equations that correspond to kinks propagatingon the Bose-Einstein condensate (BEC). These solutions are different from Gross-Pitaevskii solitonsbecause the nonlinearity of the HRMB equations arises from the interaction of the BEC and excitedatoms.I. INTRODUCTIONThe excitation and propagation of solitons in Bose-Einstein condensates (BECs) has been an active area ofstudy for a number of years. Two reviews cover the moregeneral area of BECs [1, 2] while two more put greateremphasis on soliton excitation [3, 4]. Experimental stud-ies of strongly correlated BECs have very recently be-come possible [5–8] and new phenomena have emerged[9, 10]. The fundamental parameter in these experimentsis the correlation of the BEC represented as a scatteringlength ascatt. An abrupt change of the value of ascattfrom small to large creates a stable intermediate statethat would usually have evaporated due to the stronginteractions [9].A mathematical description of this highly correlatedBEC was recently initiated by Kira [11, 12] who derivedthe so-called hyperbolic Bloch equations (HBE) to rep-resent the excited atoms of the BEC. The method is thesame as that for the semiconductor Bloch equations andis based on a cluster expansion approach of the normalcomponent of the strongly correlated BEC. The appli-cation of this method to the full BEC dynamics willnot work, because all orders in the cluster expansion arenecessary. For the excitations, this difficulty is circum-vented by the application of a non-unitary transforma-tion that uses the normal component alone by represent-ing the BEC as the vacuum state of the normal compo-nent. Then the expansion can be carried out to arbitraryorders. The first order describes singlet dynamics whichis already nonlinear and can capture interesting featuresof the dynamics. We refer the reader to [10–12] for acomplete description of this method for the BECs, andto [13] for its initial use for semiconductors in quantumoptics.Although the physics is different in these two cases, theHBE have the same structure as the semiconductor Blochequations (SBE). The only difference is a minus sign thattransforms the Bloch sphere into a hyperboloid on which∗alexis.arnaudon@ic.ac.ukthe solutions evolve. This important observation is cen-tral to our investigation of the complete integrability ofthe HBE equations following the method used for theSBE equations [12].In the case of semi-conductor optics, the SBE is cou-pled electromagnetically through a wave equation for theelectric field which contains a term dependent on the po-larisation of the semi-conductor. Recall that the HBEdoes not describes the BEC dynamics which thus requiresa coupling to the Gross-Pitaevskii equation (GP) in or-der to describe the complete dynamics. In fact, the BECwavefunction replaces the wave equation of the electricfield and the coupling is performed via a source term thatdescribes the local loss or gain of atoms in the BEC.The aim of this paper is to derive a particular approx-imation of the coupled HBE and GP equations that willbe shown to be completely integrable. This approxima-tion roughly corresponds to considering solutions of thosesystems that have small amplitude with respect to the av-erage amplitude of the BEC. In order to emphasize theparallel with optics, a similar approximation made morethan 40 years ago yielded the reduced Maxwell-Blochequations (RMB) in quantum optics, a completely inte-grable equation : see [14–17] and references therein. Theresulting equations in the present context of BECs, willbe called the hyperbolic reduced Maxwell-Bloch equa-tions (HRMB). We will also study the simplest soliton so-lution (kink) that can be observed, provided the presentapproximation remains valid.II. REVIEW OF THE RMB EQUATIONSThe Maxwell-Bloch equations appear in two differentcontexts. It first appeared in quantum optics in the con-text of the phenomenon called self-induced transparency :see for example [18, 19] for reviews on this topic. Morerecently, the quantum semi-conductor optics uses a moregeneral form for these equations that can be reduced tothe Maxwell-Bloch equations after neglecting the extrahigher order terms. We refer to [13] for a recent mono-graph on this topic. For the purpose of this work wewill prefer the semiconductor description of the Maxwell-2Bloch equations as their derivation uses the same methodthat for the derivation of the HBE equations in [12].Let us first recall the SBE in its simplest formiṖ = ω0P + (2f − 1)Ωḟ = −2 Im (ΩP ∗) .(1)P is a complex field that represents the transition am-plitude between the state of an electron and a hole. Thescalar f is the occupation number of the electrons thatvaries between −1 and 1. The complex number Ω is theRabi energy, which is proportional to the electric fieldapplied to the system. Notice that these equations con-serve the quantity η = (f− 12)2+ |P |2, which correspondsto the Bloch sphere of radius√η. These equations arealready a simplified version because they incorporate thesharp line approximation. This amounts to writing theequations with only a resonance frequency ω0 and no fre-quency averaging with a response function. These equa-tions can be derived with the cluster-expansion approach,a method similar to the BBGKY hierarchy that allowsthe computation of the many-body interactions betweenelectrons up to some order. Equations (1) only containsinglet terms where more physically realistic doublet ortriplet dynamics have been neglected.Similar interesting phenomena occur in semiconductorquantum optics where the Bloch equations (1) are cou-pled to the standard Maxwell wave equation. The resultis the semiconductor Maxwell-Bloch equations, where thewave equation for the electric field E is coupled to thepolarization P via a small material parameter α0. Thesmallness of α0 together with the use of short intensepulses allows one to neglect backscattering of waves inthe Maxwell equation. The resulting wave equation isEt + cEx = α0P , (2)where c the speed of light. We refer the reader to [14–18]and references therein for more details on the derivationof these equations.III. THE HYPERBOLIC BLOCH EQUATIONSThe hyperbolic counterpart of the Bloch equations canbe derived in the context of a strongly interacting BEC.Here we will sketch the main steps of the derivation thatcan be found in full in [12]. The successful methodof cluster-expansions can be used for BEC only in thestrongly interacting regime for the obvious reason thatthe expansion for the BEC itself will contain all orders inparticle interactions. In the strongly interacting regime,a non-negligible number of atoms are not condensed andform the normal component of the BEC. The dynamicsof these atoms can be described by a cluster-expansionwith only a few orders of interactions for relatively shorttimes after the strong correlation appears. Indeed, higherorder correlations are created from the original lower or-der ones. In order to take into account only the normalcomponent, a non-unitary transformation is applied tothe BEC wave function in order to replace the BEC bya ground state, and to concentrate only on the dynamicsof the atoms in the normal component. This techniquedeveloped in [10] together with the cluster-expansion pro-duces dynamical equations called the HBE that can con-tain various order of approximations. This equation de-rived in [12] has been shown to be more precise than theclassical Hartree-Fock approximation for strongly corre-lated BECs.In this work we will use the simplest HBE equationsthat neglect all correlations higher than the singlets. Fur-thermore we will approximate the potential describingthe inter-atomic interactions by a contact potential ; thatis, a Dirac delta function, which is equivalent to ex-pressing the Fourier transform of the contact potentialas Vk = 4π~2m−1ascatt, where ascatt is now only an effec-tive scattering length. The strongly interacting regime, ischaracterised by the limit ascatt → ∞. In practice meansthat the scattering length is saturated. This scatteringlength is experimentally controlled by the application ofan external uniform magnetic field that triggers the so-called Feshback resonance : see [12] or [1, 2] for moredetails on this interaction potential.In summary, the HBE equations that will be used inthe rest of this work is given byiṖ = ω0P + (2f + 1)Ωḟ = 2 Im(ΩP ∗) ,(3)where f is the occupation number of non-condensedatoms and P is the transition amplitude between theBEC and atoms in the normal component. ω0 is the tran-sition energy, and Ω = 4π~2m−1ascattNc is the quantum-depletion source, proportional to the number of con-densed atomsNc = Ntot − f. (4)Here, Ntot is the total number of atoms in the system,taken to be constant. Due to a sign flip in the f equation,the hyperboloid η = (f + 12)2 − |s|2 is preserved by thesolution instead of the sphere for the Bloch equations.IV. COUPLING WITH THEGROSS-PITAEVSKII EQUATIONThe next step is to couple the HBE equations (3) withthe Gross-Pitaevskii (GP) equation in order to includethe internal BEC dynamics. Recall that in the case ofthe SBE (1), the coupling with the Maxwell equation isachieved using the Rabi frequency and the electric field.In the case of BEC, the condensed atom number playsthe role of the electric field and the GP equation of thewave equation.The first approximation is the standard local-densityapproximation (LDA) which consists at studying the3BEC dynamics locally, thus neglecting the exterior trap-ping potential and use the approximation of locally ho-mogeneous BEC. The second approximation used here isto consider a one-dimensional condensate, that could stillbe valid in appropriate experiments : see for example [5].The coupling between the BEC dynamics and the HBE isimplemented as a source term in the GP equation, whichthen readsi~ψt + αψxx + βψ|ψ|2 = iβIm(P ∗)ψ, (5)where ψ is a complex-valued wavefunction, α = ~2/2m,β = 8πascattα and m the mass of a boson. First recallthat the interaction length ascatt → ∞ for strongly inter-acting BEC. Let us now write the GP equation in ampli-tude phase variables using the Madelung transformationψ(x, t) =√n(x, t) exp{iφ(x, t)} for the amplitude n(x, t)and the phase φ(x, t)nt + (2nφx)x = βIm(P∗)n (6a)φt = α((√n)xx√n− φ2x)+ 2βn. (6b)Using the LDA, we can decompose the amplitude asn(x, t) = n0+n1(x, t), where n0 is a constant backgroundwith n1 ≪ n0. The steady solution is given by n1 = 0and a time independent phase φ0(x) found by solvingαφ20,x = 2βn0. The full phase solution can thus be writ-ten asφ(x, t) = x√2βαn0 + φ1(x, t) := κx+ φ1(x, t), (7)where κ is a large number as β/α → ∞. The phaseφ0(x) is thus highly oscillating and φ1 can be consideredas a slowly varying phase. The equation (6a) for theamplitude n is then approximated at first order in κ, andtogether with the LDA, one obtains the wave equationn1,t + 8√πascattn0n1,x = ~m−14πascattn0Im(P∗). (8)For large values of n0 and ascatt, the right hand wide willcontribute more to the dynamics than the wave equation.V. THE HRMB AND RMB EQUATIONSThe system of equations derived previously is equiv-alent to the HRMB equations (10) below by using thefollowing change of variablesQ = Re(P ), P = −Im(P ),N = 2π~m(2f + 1) and E = n0 + n1.(9)The RMB and the hyperbolic HRMB equations can bewritten together asEx = PPt = EN + σ2ω0QNt = −σ1EPQt = −ω0P,(10)where we have changed frames in (8) to absorb all con-stants. We introduced σ1,2 = ±1 that selects the RMBfor σ1 = σ2 = 1 and the two possible HRMB equationsfor σ1 = −1 and σ2 = ±1. They are both integrable, asshown below, but only σ2 = 1 has been derived before.The physical interpretation for the RMB and the hRMBwith σ2 = −1 is an open problem. Notice that we haveperformed changes of independent and dependent vari-ables to simplify the form of the equations in the studyof integrability while only keeping track of the parameterω0.From (10) we see that the generalised Bloch sphere isgiven byP 2 + σ2Q2 + σ1N2 = η, (11)which is a hyperboloid when either or both of the σi arenegative.Notice that setting ω0 = 0 recovers the Sine-Gordonequation from the RMB equations and the Sinh-Gordonequation from both HRMB equations. Indeed, for theHRMB equations, the change of variables E = φx, P =sinh(φ) and Q = cosh(φ) givesφxt = 2 sinh(φ), (12)After the KdV and NLS equations [20], the Sine-Gordonequation was the next to be shown to be completely in-tegrable [21–23].VI. COMPLETE INTEGRABILITYWe will now show that all of the RMB and HRMBequations are integrable by mean of the inverse scatteringtransform (IST). Remarkably, the spectral problem asso-ciated to these equations is the Zakharov-Shabat spectralproblem [20, 23] ; that isΨx = Lσ1ΨΨt =Mσ1,σ2Ψ,(13)where Ψ = (ψ1, ψ2)T is the scattering wavefunction andL the spectral operatorLσ1 = λ[i 00 −i]+[0 E−σ1E 0], (14)for the spectral parameter λ. Well-known equations suchas the KdV or NLS equations can be written in such aspectral problem with the operator M having only pos-itive powers of λ. These would be the so-called positiveAKNS hierarchy [23]. Here, we will use the negative partof the hierarchy, where theM operator has negative pow-ers of λ. It is given for the RMB (σ1 = 1) and HRMB(σ1 = −1, σ2 = 1) byMσ1,+ =12(λ2 − ω20)(iλ[−iN Pσ1P iN]− ω0[0 Q−σ1Q 0]),(15)4whereas the HRMB or RMB case with σ2 = −1 has adifferent M operatorMσ1,− =12(λ2 − ω20)(λ− ω0)(iλ2[0 P−σ1P 0]−− λω0[0 Qσ1Q 0]+ iω20[N 00 −N]).(16)The RMB and HRMB equations appear from computingthe compatibility condition between the two equations in(13), that is∂tLσ1 − ∂xMσ1,σ1 + [Lσ1 ,Mσ1,σ2 ] = 0. (17)This allows for the use of the IST by first solving thescattering problem, i.e., compute the eigenvalues of thescattering problem with the operator L, thus evolvingthem with the M operator to finally reconstruct the so-lution by inverting the scattering problem. We will notdo this here, but leave this computation for future workbut just comment on the spectral problems. In the caseσ1 = 1, the L operator is anti-Hermitian, which meansthat the spectrum can have isolated eigenvalues in thecase of vanishing boundary conditions, i.e. E(±∞) = 0.In the hyperbolic case this operator is Hermitian, and sono discrete eigenvalues exist unless the boundary condi-tions are non-vanishing. This feature is also found in thenonlinear Schrödinger equation, where σ = 1 correspondsto the focusing NLS, and σ1 = −1 to the defocussing case.The solitons in the latter equation are of a different typethan the first and could be either dark or grey solitons,or even kinks, as in the HRMB equations – see below.An interesting feature of this spectral problem is thatalthough the RMB and HRMB equations, with eitherσ2 = ±1, share the same L operator, the M operatordiffers. Because only the operator L describes the shapeof the solitons, the shapes are uniform on the sign of σ2.The main difference between the two M operators is inthe position of the poles in the λ-plane. If σ2 = 1, thereare two simple poles λ = ±ω, and if σ2 = −1 there isa double pole at λ = ω0. Notice here that this is anarbitrary choice, and that λ = −ω0 could have also beento be the double pole. This is an unusual feature notpresent in the NLS equation which only contains positivepowers of λ in the M operator. Shifting of the zeros bysome parameter ω0 will only produce a gauge equivalentequation.VII. KINK SOLITONSAs already mentioned, soliton solutions can be derivedwith the IST method, but here we will find them by sim-ply using the travelling wave ansatz E(x, t) = E(t−c−1x)for a constant parameter c. We find the following ODEs,when using the boundary condition N(±∞) = N∞,Exx = −E(12σ1E2 + cN∞ + σ2ω2). (18)The sign in front of the E3 terms changes the type ofsolution, from sech-profile to a tanh-profile, as expected.For the solution of the RMB equations, we obtain withN∞ = −1,E(x, t) = E0 sech(12E0(t− 4E20 + 4σ2ω2x)). (19)For σ1 = −1, thus the hyperbolic case, the solution isgiven byE(x, t) = ±E∞ tanh(2E∞(t− N∞E−2∞ /2− σ2ω2x))(20)where E∞ is the value of E at x → ±∞. This solutionis often called a kink, and propagates on top of n0, theconstant background of the BEC. In the physical case,with σ1 = 1, the speed can change sign, with 0 speed ifω2E2∞= 12. In the HRMB case, with ω 6= 0, non-zeroconstants solutions can exist. Notice also the possible 0speed soliton for the RMB with σ2 = −1.ACKNOWLEDGMENTSWe acknowledge, with thanks, discussions with R. Bar-nett, M. Kira, D. Holm and A. Hone. AA acknowledgespartial support from an Imperial College London RothAward and from the European Research Council Ad-vanced Grant 267382 FCCA.[1] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari,Rev. Mod. Phys. 71, 463 (1999).[2] S. Giorgini, L. P. Pitaevskii, and S. Stringari,Rev. Mod. Phys. 80, 1215 (2008).[3] Y. S. Kivshar and B. Luther-Davies,Phys. Reports 298, 81 (1998).[4] D. J. Frantzeskakis, J. Phys. 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Integrability of the hyperbolic reduced Maxwell-Bloch equations for strongly correlated Bose-Einstein condensates

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Integrability of the hyperbolic reduced Maxwell-Bloch equations for strongly correlated Bose-Einstein condensates

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