Instabilities, nonhermiticity and exceptional points in the

A cranking harmonic oscillator model, widely used for the physics of fast rotating nuclei and Bose-Einstein condensates, is re-investigated in the context of PT-symmetry. The instability points of the model are identified as exceptional points. It is argue

Instabilities,nonhermiticityandexceptionalpointsinthecrankingmodel

W.D.Heiss1andR.G.Nazmitdinov2,3

NationalInstituteforTheoreticalPhysics,StellenboschInstituteforAdvancedStudy,andInstituteofTheoreticalPhysics,

UniversityofStellenbosch,7602Matieland,SouthAfrica2

DepartamentdeF´ sica,UniversitatdelesIllesBalears,

E-07122PalmadeMallorca,Spain3

BogoliubovLaboratoryofTheoreticalPhysics,JointInstituteforNuclearResearch,

141980Dubna,Russia(Dated:February2,2008)

Acrankingharmonicoscillatormodel,widelyusedforthephysicsoffastrotatingnucleiandBose-Einsteincondensates,isre-investigatedinthecontextofPT-symmetry.Theinstabilitypointsofthemodelareidenti edasexceptionalpoints.Itisarguedthat-eventhoughtheHamiltonianappearshermitianat rstglance-itactuallyisnothermitianwithintheregionofinstability.

PACSnumbers:03.65.Vf,03.75.Kk,02.40.Xx

1

arXiv:0709.4105v1 [quant-ph] 26 Sep 2007

Quantuminstabilitiesareattractingconsiderableat-tentioninavarietyofphysicalsituations.TheycanbeassociatedwiththeformationofsolitonsandvorticesinBose-Einsteincondensateswithasuddenchangeofthemomentofinertiaofarotatingnucleus(see,forexam-ple,Ref.2andreferencestherein)andatransitionfromone-totwo-dimensionalnuclearrotationAparticularexampleofinterestistheHamiltonian

= ω1(a a1+1H1

)+

2

+i g1(a 1a2 a2a1) i g2(a1a2 a2a1)(1)

usedincondensedmatterphysicstodescribeinasim-plewaytheinteractionbetweenanatomandaradiative

eld[4].Notethatthebi-linearformof(1)correspondstoalinearisedversionofsomemoregeneralinteractions.Asdiscussedbelowthismaybringaboutaninstability.Higherordertermsmayormaynotremovesuchinsta-bility.

Usingthestandardrelations( =m=1)

ak=pk(2)xk+i

22ωk

a k=

2xk i

2ωk

pk

(3)

wherex1=x,x2=y,andchoosingspecialvaluesfor

√

thestrengthconstantsg1= (ω1+ω2)/2

ω1ω2,onerecognisesthewell-known

crankingHamiltonian(Routhian)

H=

p2x

2x+

2

p2y

2

y2 Lz

(4)

whichhasbeenappliedinnuclearphysicsand

forrotatingBose-Einsteincondensates(cfTheHamiltonian(4)appearsasthesumofhermitianopera-torsandisexpected-naivelyat rstglance-tobeitself

ahermitianoperator.Thesameholdswheniswrit-teninsecondquantisedformInthefollowingweexploretheformalcharacteroftheinstabilitypointsof

.WearguethattheoperatorsarenolongerHandH

hermitianatthesepoints,infact,weshow,thatthesepointsareexceptionalpoints(EP)Non-hermitianHamiltonoperatorshaveattractedwidelyspreadinterestduringtherecentyears(see[10]),beitinthecontextofe ectivetheories[11],orinthecontextof ndingaequivalent[12]orinthecontextofPT-symmetry(PTistheproductoftheparityandtimereversaloperator).Onespeci caspectofnon-hermitianoperatorsaretheEPs,beingsingularitiesofspectrumandeigenfunctions.Assuch,theyareusu-allyofparticularphysicalsigni cance.Theyhavebeendiscussedinagreatvarietyofphysicalapplications:inoptics[14],inmechanics[15],ascoalescingresonances[16,17],inatomicphysics[18],andinmoretheoreticalcontextinPT-symmetricmodelsorinconsideringtheirmutualin uencetonamejustafew.Initssim-plestcasetheygiverisetolevelrepulsionbeingthemorepronouncedthenearertheylietotherealaxis.Depend-ingontheparticularsituationtheycansignalaphasetransitionInthepresentcasetheEPisassociatedwiththeonsetofaninstability.NotethattheHamilto-nian(4)issymmetricunderPT-operationirrespectiveofaspecialchoiceofparameters( → underT)ItiswellknownthataBogoliubovtransformationoftheHamiltonian q+ax q =B ay (5) q a y

a q+yyieldstheform(cf[6])

=ω+(q q++1H+

2)

(6)

A cranking harmonic oscillator model, widely used for the physics of fast rotating nuclei and Bose-Einstein condensates, is re-investigated in the context of PT-symmetry. The instability points of the model are identified as exceptional points. It is argue

withtheeigenmodeenergies

ω2±=

1

(ω2x ω2y)2+8 2(ω2x+ω2y

)(7)

.Itisalsoknown[6,22]thatω2

becomesnegativewhentherotationalspeed liesbetweenmin(ωx,ωy)andmax(ωx,ωy).Inthefollowingweassumethatωx>ωy.Atthepointswherethetwoeigenmodesvanish,thatiswhenω ,1=+ω andω ,2= ω coalesce,thematrixBpointsin(5) cbecomes1=ωyorsingular. c2=ωxThissignallinghappensanatinstability.thecriticalThecoalescenceisreminiscentofthebehaviourofanEP.Tocon rmthatweareinfactencounteringagenuineEPandnotausualdegeneracy,wehavetoanalysetheeigenfunctionsoftherespectiveHamiltonians.Ofcourse,thisiscloselyrelatedtothesingularbehaviouroftheBogoliubovtransformationB.

Toilluminateboth,theunderlyingphysicsandthemathematicalstructure,itisconvenienttoconstructthematrixUconnectingtheoriginalcanonicalcoordinates p

and rwiththequasi-bosonoperatorsqkandq

k,thatis

px

pyx

q+y

=U q q q +

.(8)Asa rststepweaimmalmodecoordinatesP

atthegeneralisedclassicalnor-=(P+,P )andX =(X+,X )inwhichHassumestheform

2H =

P+2

X+

2

+

P

2X 22

.

(9)

Thisisachievedbysolvingtheclassicalequationsofmo-tion

d

r(10)k

d

p(11)

k

whichcanbewritteninmatrixform

d

100

01 02 0 ω20

00

x

.(14)

ω2y

2

Notethat(4)canbewrittenas

H=

p r

H p

r

.

(15)

Thesolutionof(12)isobtainedbyexponentiationand

reads

p (t) r(t) =Uexp(Dt)V p (0)

r(0) (16)whereD=diag( iω+, iω ,iω ,iω+)isthediagonal

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