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DIPARTIMENTO DI SCIENZE FISICHE ED ASTRONOMICHE PALERMO, ITALY . Casimir-Polder force between two uniformly accelerating atoms and the Unruh effect. Quantum Field Theory under external conditions (QFEXT09), Norman, September 2009. J. Marino and R. Passante.
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DIPARTIMENTO DI SCIENZE FISICHE ED ASTRONOMICHE PALERMO, ITALY Casimir-Polderforcebetweentwouniformlyacceleratingatoms and the Unruh effect Quantum FieldTheory under externalconditions (QFEXT09), Norman, September 2009 J. Marino and R. Passante Dipartimento di Scienze Fisiche ed Astronomiche, Palermo, Italy
In thistalk… …Iwilldiscuss the Casimir-Polderforcebetweentwoacceleratingatomsfor • smallaccelerations • high accelerations atomictransitionfrequency I will show how Unruh Effectchanges the scalingwith R of the interatomicenergy
The Unruh effect Rindler coordinate transformation Rindlerspacetimeinterval
Expectationvaluesofparticlenumbers ActionofMinkowskioperators on Minkowskivacuum Actionof right Rindleroperators on Minkowskivacuum An acceleratedobserverassociates a thermalbathoffictiousparticlesto the Minkowskivacuum
Acceleratedhydrogenatom and the Unruh effect in QED MultipolarGauge Linear couplingbetweenfield and atom propertimeof the atom We separate vacuumfluctuations and reactionradiationcontributionsto the energylevelshifts R. Passante, Phys. Rev. A, 57, 3 (1998)
Vacuumfluctuationscontributionto radiative levelshifts • Unruh effect can bedetectedforatomicaccelerations • The termisrelevant at accelerationsof the sameorder • Thereisno equivalencebetween scalar Unruh effect and QED Unruh effect
LambShiftof a uniformlyacceleratedatom near a conductingplate (1) • Scalar Case • Forthere are onlysmallcorrectionsto the staticatom-plate Casimir force. • Forthe LambShiftisreminescentof the LambShiftofaninertialatom, in itsexcited state, in frontof a plate L. Rizzuto Phys. Rev. A 76, 062114 (2007)
LambShiftof a uniformlyacceleratedatom near a conductingplate (2) • EM Case • Westillrequire high atomicaccelerationtodetect Unruh effect. • We can useanisotropiesofatomicpolarizabilitiestoweaken or increase the atom-wallforce (for high atomicaccelerations). QED vacuum L. Rizzuto, S. Spagnolo Phys. Rev. A 79, 062110 (2009)
Ourphysicalsystem… The twoatomshave the sameacceleration The twoatomssee the sameUnruh bath
Goedecke’s method Casimir-Polderinteraction at finite temperature Weconsider the T=0 interactionpotential For a thermalbathwehave . Goedeckesubstitues the averageparticlenumber in the T=0 - interactionpotential G. H. Goedecke, R. C. Wood, Phys. Rev. A 60, 3 (1999)
Casimir-Polderinteraction at finite temperature In the high temperature limit In the low temperature limit Near zone Far zone
We compare the radiative vacuumshiftofanacceleratedhydrogenatom with the radiative vacuumshiftofanhydrogenatom in a thermalbath Fromthiscomparison, weguess the expression for the average mode numberforanacceleratedatom
withthisaverageparticlenumber, we can write the Casimir-Polderpotentialbetweentwoacceleratedatoms : negligibletransitionprobabilitytoexcitedstates
Extensionof V(R) to the compexplane Weperform a rotation of the integrationline in the first quadrantof the complexplane Poles on the immaginaryaxis:
Near zone Far zone • In far zone the Unruh effectintroduces a correctionto the interactionenergy • The accelerationdependentterms are in bothcasescomparableto the stationariesoneswhen
Thisresultholdsboth in near and in far zone; • itisrelevantonlyto high accelerations; • itissimilarto high temperature limitofthe Casimir-Polderinteraction at finite temperature
Casimir-Polderforcebetweentwouniformlyacceleratingatoms in the highaccelerationlimit Spontaneousexcitation Inertialintermolecularpotentialforexcitedmolecules E. A. Power, T. Thirunamachandran, 48, 6 (1993). Resonancefrequencyof the atomictransition Resonantcontributionofaverage mode number
For a characteristicatomictransition, and high accelerations are againrequired. Reminescentof the exchangeofrealphotonswhichoccurswhen the twomolecules are excited. Far zone
Foundationsof the method Non-relativisticdynamics: Under theseapproximationswe can use the staticpotentialand the averageparticlenumber in the sameexpression
Conclusions • The Casimir-Polderforcebetweentwoacceleratedatomscontainstermsderivingfrom the Unruh effect, relevantonly at . • In far zone theUnruh effectintroduces a term, thatscaleswith the fifthpowerof R. • Usinganheuristicmethod, wefindforanpowerlaw in far zone.
Future developments Actually, we are workingto a IV ordermethodthatseparatesvacuumfluctuations and reactionradiationcontributions in the interactionenergyof the twomolecules. Thisapproachhas the advantageto solve from first principles the problemofCasimir-Polderforcebetweentwouniformlyacceleratingatoms. Acknowledgments This talk hasbeensupportedby a grant from the International AdvisoryCommitteeof the QFEXT09