Non-thermal effects of acceleration in the resonance interaction between two uniformly accelerated atoms

0.2 MB·

by  Lucia Rizzuto

Non-thermal effects of acceleration in the resonance interaction between two
uniformly accelerated atoms
Lucia Rizzuto1,2, Margherita Lattuca1, Jamir Marino3,4, Antonio Noto1,5,
Salvatore Spagnolo1, Wenting Zhou1,6, and Roberto Passante1,2
1 Dipartimento di Fisica e Chimica, Universita` degli Studi di Palermo and CNISM, Via Archirafi 36, I-90123 Palermo, Italy∗
2 INFN, Laboratori Nazionali del Sud, I-95123 Catania, Italy
3 Institute of Theoretical Physics, TU Dresden, D-01062 Dresden, Germany
4 Institute of Theoretical Physics, University of Cologne, D-50937 Cologne, Germany
5 Laboratoire Charles Coulomb UMR 5221 CNRS-UM2, D´epartement Physique Th´eorique,
Universit´e Montpellier 2, F-34095, Montpellier Cedex 5, France and
6 Center for Nonlinear Science and Department of Physics,
6
Ningbo University, Ningbo, Zhejiang, 315211, China
1
0
We study the resonance interaction between two uniformly accelerated identical atoms, one ex-
2
citedandtheotherinthegroundstate,preparedinacorrelated(symmetricorantisymmetric)state

g and interacting with the scalar field or the electromagnetic field in the vacuum state. In this case
u (resonance interaction), the interatomic interaction is a second-order effect in the atom-field cou-
A pling. Weseparatethecontributionsofvacuumfluctuationsandradiationreactiontotheresonance
energyshift of thesystem,andshow thatonlyradiation reaction contributes,while Unruhthermal
2
fluctuationsdonotaffecttheresonanceinteraction. Wealsofindthatbeyondacharacteristiclength

scale related tothe atomic acceleration, non-thermal-likeeffects in theradiation reaction contribu-
]
h tion change the distance-dependence of the resonance interaction. Finally, we find that previously
p unidentified features appear, compared with the scalar field case, when the interaction with the
- electromagnetic fieldisconsidered, asaconsequenceofthepeculiarnatureofthevacuumquantum
t noise of theelectromagnetic field in a relativistically accelerated background.
n
a
u PACSnumbers: 12.20.Ds,03.70.+k,42.50.Ct
q
[
I. INTRODUCTION tions between two uniformly accelerated atoms could be
4
verypromisingcandidatesfordetectingtheUnruheffect,
v
even with reasonable values of the acceleration [14, 15].
2
Astrikingconsequenceofquantumfieldtheoryinnon-
0 Among the large number of open fundamental chal-
Minkowskian spacetimes is the observer-dependent par-
5 lenges in this field, a long-standing question is whether
4 ticle content of the quantum vacuum. An archetypi- or not the effect of a relativistic acceleration is strictly
0 cal manifestation of this feature is the Unruh effect: a
equivalenttoathermalfield[16,17]. Ithasbeenrecently
. uniformly accelerated observer in the Minkowski vac-
1 shown,forexample,thatnon-thermalfeaturesassociated
0 uum perceives vacuum fluctuations as a thermal field, withuniformaccelerationmanifestintheradiativeprop-
6 with a temperature TU proportional to its acceleration, erties of single accelerated atoms [15, 18–22]. Also, re-
1 T = ~a [1–3]. The Unruh effect has stimulated in-
U 2πkBc centworksonentanglementgenerationorCasimir-Polder
: tense theoretical investigations in last 40 years, includ-
v interactions between uniformly accelerated atoms, have
i ing connections with cosmology [4, 5] and applications shown that non-thermal effects of acceleration arise in a
X
to quantum optics and quantum information [6, 7]. Ex-
system of two or many particles [17, 23, 24].
r perimentalproposalsfor its measurementusing particles
a moving in circular accelerators [8], or electrons acceler- In this context, it has been recently shown that the
ated by ultrastrong laser fields [9], as well as in ana- Casimir-Polder (CP) force between two uniformly accel-
log models of condensed matter physics [10], have also erating atoms in their ground state exhibits a cross-over
been suggested. However, despite these intense efforts fromashort-distancethermalbehaviortoalong-distance
the problem of detecting the Unruh effect remains open non thermal behavior,with respect to a reference length
because of the very high accelerations, of the order of identified with za = ca2, where a is the proper accelera-
1020m/s2, necessary to obtain a Unruh temperature tion of the atoms. This characteristic length scale coin-
∼
of a few Kelvins. On the other hand, a direct verifi- cideswiththebreakdownofalocalapproximatedescrip-
cation of the Unruh effect could allow a deeper under- tion of the two-body system in terms of a Minkowskian
standing of some controversies about its interpretation space-time [17]. Indeed, Casimir-Polder forces between
[11–13]. In this direction, it has been recently argued neutralatoms arise fromthe retardedinteractionamong
that interatomic van der Waals/Casimir-Polder interac- thedipolesinducedandcorrelatedbythezero-pointfield
fluctuations, and the field quanta mediating the inter-
action between the two atoms are affected by the non-
inertialcharacterofrelativisticacceleration,atlargedis-
∗Electronicaddress: [email protected] tances [14, 17].
2
In this paper we investigate the resonance interaction specifically, a different scaling of the interaction energy
between two uniformly accelerated identical atoms, one with the distance and a dependence on the acceleration,
excited and the other in the ground state, prepared in a when compared to the “Unruh-thermal” Casimir-Polder
correlatedstate (symmetric or antisymmetric) and show interaction case [17]. Following the procedure adopted
that it exhibits a pure non-thermal behavior, carrying in [36–39], we separate, at second order in perturbation
no signature of Unruh thermal fluctuations on the in- theory, the contributions of vacuum fluctuations and ra-
teratomic force. Nevertheless, the relativistic acceler- diation reaction to the resonance interaction energy be-
ation still causes a qualitative change of the distance- tween the two atoms. We show, both for the scalar and
dependence of the interaction between the two atoms. electromagnetic cases, that the resonance interaction is
Resonanceinteractionsbetweenatomsoccurwhenone relatedonlytotheradiation-reactioncontribution. Thus,
ormoreatomsareintheirexcitedstateandanexchange Unruh thermal fluctuations do not affect the resonance
of real photons is involved [25, 26]. If the two atoms are interatomicinteraction. Also,weshowthatinthelimitof
prepared in a factorized state, resonance Casimir-Polder large accelerations, the distance dependence of the reso-
interactions require a fourth-order perturbation theory. nanceinteractionchangesqualitatively. Thusourresults
In this case, the interaction scales as R−2 for large in- permit one to highlight non-thermal signatures of the
teratomic separations, R λ (R being the interatomic atomic acceleration through the second-order resonance
distance and λ the main≫wavelength associated to the interaction between identical atoms.
atomictransitions). Theseeffectshavebeenrecentlydis- The paper is organized as follows. In Sec. II we in-
cussed in the literature, in particular in connection with troduce our model and discuss the resonance interaction
the relevant problem of the Casimir-Polder interaction between two correlated accelerated atoms interacting
between two nonidentical atoms when one of them is in with the scalar field in the vacuum state. In Sec. III
an excited state [27–30]. On the other hand, resonance we generalize our procedure to the more realistic case of
interactions canalso occur when two identicalatoms are accelerated atoms interacting with the electromagnetic
prepared in a correlated (symmetric or antisymmetric) field. Finally, Sec. IV is devoted to our conclusions and
state,andinthiscasetheymanifestasasecond-orderef- perspectives. Details of some calculations are given in
fectintheelectriccharge. SuchinteractionsscaleasR−1 the Appendix.
in the far zone,and thus they are of a longer rangecom-
pared with dispersion interactions. Recently, the possi-
bility to enhance resonance forces between atoms placed
in nano-structured materials such as a photonic crystal
has been discussed [31]. Also, such effects have been in-
II. RESONANCE INTERACTION ENERGY
vestigatedin relationto the resonantenergytransferbe- BETWEEN ACCELERATED ATOMS: THE
tween molecules, and it has been argued that they could SCALAR FIELD CASE
play a fundamental role in some biologicalcoherent pro-
cesses [32–34].
Letus considertwoidentical atoms(labelled as Aand
Asmentioned,inthispaperweshallconsidertwoiden-
B) modeled as point-like systems with two internal en-
tical atoms prepared in a correlated (symmetric or anti- ergy levels, 1~ω , associated with the eigenstates g
symmetric) state and uniformly accelerating in vacuum, ∓2 0 | i
and e , respectively, and separated by a distance z.
and investigate the resonance interaction between them. | i
We assume that the two atoms are acceleratingwith the
The radiative properties of uniformly accelerated atoms
sameuniformaccelerationalongtwoparalleltrajectories,
prepared in a maximally entangled state, have been in-
x (τ) and x (τ) (therefore their distance is constant),
A B
vestigated in the literature until very recently [23, 35],
and interacting locally with a real massless scalar field
but here their resonance interaction is discussed. A
in its vacuum state. Also, as usual, we suppose that ω
0
motivation of our work is to explore situations where
includes any direct modification of the atomic transition
the effects of the acceleration could be exclusively non-
frequency due to the accelerated motion. The Hamilto-
thermal and manifest sharp differences with respect to
nian describing the atom-field interacting system in the
Unruhthermal-likeeffects. Sincetheatomsareprepared
instantaneous inertial frame of the two atoms is (τ is a
in a correlated Bell-type state, our calculation requires
proper time) [38, 39]
only second-order perturbation theory. Also, since reso-
nanceinteractionsaremuchmoreintensethandispersion
dt
Cdiadsaitmeirf-oProoldbesrerivnitnegratchtieonesff,etchteyofcoauccldelbereaatesduimtaobtlieocnanin- H(τ)=~ω0σ3A(τ)+~ω0σ3B(τ)+ ~ωka†kakdτ
k
quantum field theory. X
+λ(σA(τ)φ(x (τ))+σB(τ)φ(x (τ))), (1)
2 A 2 B
We firstconsiderthe atomsinteractingwiththe scalar
field andthenwe generalizeour investigationto the case
of the electromagnetic field. We show that specific fea- where σA(B) (i=1,2,3)are pseudospin operatorsin the
i
tures, not present for inertial atoms, appear in the res- Hilbertspaceofthe internaldegreesoffreedomofatoms
onance interaction as a consequence of the acceleration; A and B, and a†k, ak are the creation and annihilation
3
bosonic operators of the scalar field observable. The method consists of rewriting these two
different contributions in terms of two effective Hamilto-
φ(x,t)= 2V~ω ak(t)eik·x+a†k(t)e−ik·x . (2) bnuiatniosn, HtovefftfheanrdesHonsearfnft, aenndertghyenshcifotm. pTuhtiisngapthperioraccohnhtrais-
k r k
X h i been used to investigate the effect of the atomic acceler-
We wantto calculate the resonanceenergyshift ofthe ationon the radiativeproperties ofsingle atoms [15, 18–
systemofthetwoacceleratedatoms,duetotheirinterac- 22] and has been recently generalized to calculate the
tion with the scalarfield. As it is known,the interaction Casimir-Polderinteraction between two uniformly accel-
between atoms can be interpreted in terms of radiation erated atoms [17].
source fields, as well as of vacuum field fluctuations [40].
In this paper, we obtain the resonance interaction fol-
lowing a procedure proposed in [36–39], which consists We first derive the effective Hamiltonians Hvefff and
of separating at a given order in perturbation theory, Heff at second order in the atom-field coupling. After
sr
the contributions of vacuum fluctuations (vf) and self- somealgebra(detailsaregivenintheAppendix),wefind
reaction (sr) to the time evolution of a generic atomic the following expressions
λ2 τ
Heff = i dτ′CF(x (τ),x (τ′)) σf (τ),σf (τ′) , (3)
(cid:16) A (cid:17)vf − 2~Zτ0 A A h 2,A 2,A i
and
λ2 τ τ
Heff = i dτ′χF(x (τ),x (τ′)) σf (τ),σf (τ′) + dτ′χF(x (τ),x (τ′)) σf (τ),σf (τ′) (4),
(cid:16) A (cid:17)sr − 2~)Zτ0 A A n 2,A 2,A o Zτ0 A B n 2,A 2,B o)
wherewehaveintroducedthestatisticalfunctionsforthe called a subradiant state because its spontaneous decay
scalarfieldgivenintheAppendix. Analogousexpressions is inhibited. These two correlated states are degenerate,
for (Heff) are obtained by exchange of A and B in andthusperturbationtheoryfordegeneratestatesshould
B vf/sr
the relations above. beused,inprinciple. However,itispossibletoshowthat
The resonance interaction between two atoms moving if the dipole matrix elements of the two identical atoms
on the stationary trajectories x (τ) and x (τ), is then are equal in modulus and the photon dispersion relation
A B
obtainedbyevaluatingtheexpectationvalueoftheeffec- issymmetricbetweenkand k,thesymmetricandanti-
−
tive Hamiltonians Heff = (Heff) +(Heff) , symmetric subspaces do not mix each other at the order
vf/sr A vf/sr B vf/sr
considered. Then, in this case the degenerate symmetric
onthe correlatedstateofthe twoatoms,andtakinginto
and antisymmetric states can be treated independently
account only the terms depending on the atomic separa-
and the second-order energy shift is the same as in the
tion.
nondegenerate perturbation theory for both states. Dif-
Inordertodothat,wesupposethetwoatomsprepared
ferent methods to obtain super- and subradiant states
in one of the correlatedstates
for two-level systems, have been proposed (see, for ex-
1 ample,[42, 43]). Onthe other hand,it hasbeen recently
ψ = ( g ,e e ,g ) , (5)
| ±i √2 | A Bi±| A Bi shown that entanglement between accelerated systems
can be induced by the Unruh bath [23, 24]. To obtain
where, as mentioned before, g (e) indicates the ground the resonance energy shift for the system considered, we
(excited) state of the atom. In these states the atomic now evaluate the expectation values of Heff and Heff
vf sr
excitationisdelocalizedamongthetwoatoms. Thesym- on state (5) (symmetrization with respect to A ⇆ B is
metrical state is called a super-radiant state because in
necessary to obtain the total energy shift). After some
the Dicke model its decay rate is larger than that of the
algebra, we get
individualatoms,yieldingacollectivespontaneousdecay
[41]. Onthecontrary,theantisymmetricsuperpositionis
4
iλ2 τ
(δE) =(δE ) +(δE ) = dτ′CF(x (τ),x (τ′))χ (τ,τ′)+(A⇆Bterms), (6)
vf A vf B vf − ~ A A A
Zτ0
iλ2 τ
(δE) =(δE ) +(δE ) = dτ′χF(x (τ),x (τ′))C (τ,τ′)
sr A sr B sr − ~ A A A
Zτ0
iλ2 τ
dτ′χF(x (τ),x (τ′))C (τ,τ′)+(A⇆Bterms), (7)
− ~ A B A,B
Zτ0
wherewehaveintroducedtheatomicstatisticalfunctions field). This peculiarity has relevant consequences when
we consider the resonance interaction between acceler-
1
C (τ,τ′)= ψ σf (τ),σf (τ′) ψ , (8) ated atoms; in fact, we will now show that resonance
A,B 2h ±| 2,A 2,B | ±i interactions do not display any signature of the Unruh
1 n o
χ (τ,τ′)= ψ σf (τ),σf (τ′) ψ . (9) thermal effect (which is related to the correlations of
A,B 2h ±| 2,A 2,B | ±i vacuum field fluctuations in the locally inertial frame).
h i
Nevertheless, the atomic acceleration will cause a quali-
Thevacuumfluctuationcontribution(6)givesonlythe
tative change of the interaction between the two atoms.
Lambshiftofeachatom,asiftheotheratomwereabsent,
This situation should be compared with the case of the
and thus it does not contribute to the interatomic inter-
Casimir-Polder dispersion interaction between two un-
action energy. Similarly, the first term in (7) describes
correlated atoms, where atomic dipoles are induced and
the self-reaction contribution to the Lamb shift of each
correlated by the zero-point fluctuations of the field and
atom. On the contrary, the second term in (7) is the
the interaction is directly related with the spatial corre-
relevant one for the resonance interaction; it describes
lations of vacuum fluctuations.
the interactionof one atom with its own emitted field as
The procedure outlined above is general and valid for
modifiedbythepresenceoftheotheratom(asexpressed
by χF(x (τ),x (τ′))). It depends on the distance be- any arbitrary stationary trajectory. We now focus on
A B
the case of two atoms moving with the same uniform
tween the two atoms, and it is the only contribution to
acceleration along the trajectory
the interatomic interaction at the order considered.
The result above shows that the resonanceinteraction
is due to the radiation reaction contribution δE only.
sr c aτ c2 aτ
Thisisindeedexpectedfromaphysicalground. Theres- t(τ)= sinh , x (τ)= cosh ,
onance interaction originates from the exchange of one a c A/B a c
photon between the two atoms that are in a correlated yA/B(τ)=0 zA(τ)=zA ,zB(τ)=zB, (10)
state; thus, contrary to the dispersion interaction where
theatomiccorrelationisinducedbyvacuumfluctuations, We first evaluate the linear susceptibility of the scalar
itisnotrelatedtononlocalvacuumfieldcorrelations,but field (A.12) and the atomic correlation function (8). We
is entirely due to the field radiated by the atoms (source have
χF(x (τ),x (τ′))= ~ 1 ∞dωg(ω,z,a)(eiω(τ−τ′) e−iω(τ−τ′)), (11)
A B −8π2c2z N(z,a) −
Z0
p
and The upper (lower) sign in (12) refers to the symmetric
(antisymmetric) superposition in (5), respectively.
CA,B(τ′,τ)= 1(eiω0(τ−τ′)+e−iω0(τ−τ′)),(12) Now, substituting (11) and (12) in Eq. (7) and taking
±8
into account only z dependent terms, we finally obtain
−
where we have defined N(z,a) = 1 + (za/2c2)2 and the resonance interaction between the two accelerated
g(ω,z,a) = sin(2ωcsinh−1(za)). Finally, z = z z . atoms
a 2c2 B − A
5
λ2 1 ∞ 1 1
δE =(δE ) +(δE ) = dωg(ω,z,a) + . (13)
A sr B sr ∓16π2c2z N(z,a)Z0 (cid:16)ω+ω0 ω−ω0(cid:17)
p
We have also taken the limits τ and τ in wascharacterizedbyaz−4powerlawdecay,fordistances
0
→ −∞ → ∞
the integral above, since we are considering the atoms z c2/a.
≫
uniformly accelerating in vacuum for all times. We also stress that Eq. (15) exhibits a global over-
The integral in the equation above can be evaluated all pre-factor depending on the inverse of the acceler-
exactly, giving ation, while the “thermal-Unruh” analogy would have
suggested the presence of a Unruh term with tempera-
λ2 1 2ω c za
δE = cos 0 sinh−1( )(14) tureTU =a/2π,directlyproportionaltotheacceleration
∓16πc2z N(z,a) a 2c2 [17]. Thus, our result shows that it is possible to single
(cid:16) (cid:17)
out metric effects associated to relativistic accelerations
Equation(14)isthempainresultofthissection. Sincethe from the usual “Unruh thermal-like” effects.
interaction energy is entirely due to the radiation reac- The limit z z = c2/a in (16) gives back the res-
a
tion contribution, the atomic accelerationdoes not yield onance interact≪ion for the inertial case. For typical in-
thermal signatures in the resonance interaction; its only teratomic distances z 10−6m, this limit is valid also
effect is enclosedin the normalizationfactor N(z,a)and for considerable accele∼rations, thus suggesting that the
in the function g(ω,z,a). More precisely, a comparison resonance interaction is almost insensitive to the atomic
with the Casimir-Polder interaction between two accel- accelerationwhen z z . Actually, sucha behaviorcan
a
erated atoms as in [17] shows that while in that case the be expected from th≪e following physical considerations.
effectofaccelerationisathermalcorrectionwiththeUn- Resonance interactions arise from the exchange of real
ruh temperature T = ~a/2πckB (due to the presence of photons between the atoms. If the distance between the
a factor coth(πcω/a) in the vacuum fluctuations contri- atomsis muchsmallerthanz , duringthe time takenby
a
bution), in the present case (resonance interaction) the the photonemittedbyoneatomto reachthe otheratom
effects of atomic accelerationare not in the form of such (t z/c), the accelerating atoms move of a distance x
a thermal term. Most importantly, the presence of the wh∼ich is much smaller than their interatomic distance z;
factorN(z,a)yieldsachangeofthescalingoftheinterac- thusthephotonmediatingtheinteractioncannotdiscern
tion with the distance. Indeed, we canidentify a charac- the atomic motion, and the interaction is essentially the
teristiclengthscale,za =c2/a,inanalogytotheresultof same between stationary atoms.
[17] for the dispersion interaction. For distances smaller Finally, it is worth noting from (16) that the iner-
thanza, it is possible to finda localinertialframe where tial resonance interaction decreases as z−1 for any inter-
thelinearsusceptibilityoffieldisfairlywelldescribedby
atomic distance. This is a peculiarity of the scalar field
its static counterpart; onthe other hand, signals spread- model we have considered. The distance behavior is dif-
ing over distances larger than z may be affected by the
a ferent in the case of the electromagnetic field, where we
non-inertial character of relativistic acceleration. Ac- candistinguisha near anda far zone, as we shalldiscuss
cordingly, we expect that relativistic accelerations can
in the next section.
deeply modify the qualitative behavior of the resonance-
interactionenergy,inparticularits distance dependence.
In fact, in the limit z c2/a, we get
III. THE ELECTROMAGNETIC FIELD CASE
≫
λ2 1 2ω c za
0
δE cos ln , (15) We now extend our investigation to the case of two
≃∓8πz2a a c2
(cid:16) (cid:16) (cid:17)(cid:17) uniformlyacceleratedatomsinteractingwiththeelectro-
while for z c2/a, we recover the inertial result magnetic field in the vacuum state.
≪
To describe our system, we adopt the Hamiltonian in
λ2 1 ω z
0 themultipolarcouplingschemeandinthedipoleapprox-
δE cos . (16)
≃∓16πc2z c imation
(cid:16) (cid:17)
Thus, the resonance interaction strongly bears signa- dt
tures of a relativistic acceleration, resulting in a differ- H =HA+HB + ~ωka†kjakjdτ
ent power-law distance dependence, scaling at large dis- kj
X
tances as z−2 rather than the usual z−1 of the inertial µ (τ) E(x (τ)) µ (τ) E(x (τ)), (17)
A A B B
− · − ·
case (16). This result should be compared with that ob-
tainedin[17],whereitwasshownthat,asaconsequence wherej=1,2isthepolarizationindex,E(x(τ))istheelec-
of metric effects, the scalar Casimir-Polder interaction tric field operator, and µ = er is the atomic dipole mo-
between two uniformly accelerated ground-state atoms, ment operator. As in the previous section, the atoms,
6
withtransitionfrequencyω ,haveauniformacceleration the direction of acceleration, x. A simple calculation
0
a along the x direction and are separated by a constant shows that the only nonzero components of are
ℓm
G
distance z along the z direction, while x (τ) are the the xx, yy, zz, xz, and zx components. In particular,
A/B
trajectories of the two atoms. (x (τ),x (τ′)) = (x (τ),x (τ′)) (for ℓ = m);
ℓℓ A B mm A B
G 6 G 6
As discussedbefore, the resonantinteractionenergy is therefore,the Rindler noise evaluated onthe atomic tra-
related only to the radiation reaction contribution and jectoriesofthetwoacceleratedatoms,isnotisotropicand
is obtained from the expectation value of the effective displaysanon-diagonalcomponent. Asimilaranisotropy
Hamiltonian (Heff) +(Heff) on the state ψ is not present in the case of a single uniformly acceler-
A sr B sr | ±i
ated atom in the unbounded space, where it is possible
e2 τ to show that the Rindler noise is isotropic. Actually, in
δE = dτ′χF (x (τ),x (τ′))CA/B(τ,τ′)
− 2 ℓm A B ℓm our system we have two characteristic spatial directions,
Zτ0 namelythe directionoftheaccelerationandthedistance
+(A⇆Bterms). (18)
between the atoms; in this sense, the anisotropic aspect
of the Rindler function can be ascribed to the spatially
In order to evaluate this quantity, we first obtain the
extendedcharacterofthetwo-particlesystemconsidered.
statisticalfunctionsforthefieldandtheatoms. Thesus-
ceptibility of the electromagnetic field in the accelerated From Eq. (19), we can obtain the linear susceptibility
frame can be obtained from the two-point field corre- oftheelectromagneticfieldintheproperreferenceframe,
lation function in the proper reference frame of the two
acceleratedatoms(Rindlernoise)[44]. Afterlengthycal-
culations involving Lorentz transformations of the elec- χF (x (τ),x (τ′))= i 0[E (x (τ)),E (x (τ′))](200).
tromagnetic field, we obtain ℓm A B ~h | ℓ A m B | i
Itsexpression,intheformofanintegraloverfrequencies,
(x (τ),x (τ′))= 0E (x (τ))E (x (τ′))0 is
ℓm A B ℓ A m B
G h | | i
~a4 1
=
4πc7(sinh2 a(τ−2τc′−iǫ) −(2zca2)2)3 χ (x (τ),x (τ))= 1 ∞dω(eiωu e−iωu)
[δ za(n q n q )]sinh2 a(τ −τ′) ℓm A B 2πz3 Z0 −
×( ℓm− c2 m ℓ− ℓ m 2c az
f (a,z,ω)+ F (a,z,ω) cos(ωS)
+(za)2[δ 2n n ] ×"(cid:18) ℓm 2c2 ℓm (cid:19)
2c2 ℓm− ℓ m
az
× 1+2(δℓm−qℓqm)sinh2 a(τ2−cτ′)#) (19) +(cid:18)gℓm(a,z,ω)+ 2c2Gℓ,m(a,z,ω)(cid:19)sin(ωS)#, (21)
(cid:20)
(ℓ,m = x,y,z). n = (0,0,1) is the unit vector along where u=τ τ′ and S = 2csinh−1 za . We have also
− a 2c2
the z direction and q = (1,0,0) is the unit vector along defined the functions
(cid:0) (cid:1)
1 ωz a2z2 a2z2
f (a,z,ω) = (δ 3n n )+ 2(δ +q q n n )+(δ q q 2n n ) 1+ (2,2)
ℓm N2 c " ℓm− ℓ m 4c4 (cid:18) ℓm ℓ m− ℓ m ℓm− ℓ m− ℓ m (cid:18) 2c4 (cid:19)(cid:19)#
1 a2z2 a2z2 a2z2 a2z2
g (a,z,ω)= δ 1+ +q q 1+ 3n n 1+
ℓm −N5/2 ℓm 4c4 ℓ m 4c4 c4 − ℓ m 2c4
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
ω2z2 a2z2 a2z2 a2z2
+ δ 1+ q q n n 1+ , (23)
c2N3/2 ℓm 4c4 − ℓ m 4c4 − ℓ m 2c4
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
ωz a2z2 The functions f (a,z,ω) and g (a,z,ω) are non-
F (a,z,ω)= n q n q 1 (2,4) ℓm ℓm
ℓm m ℓ− ℓ m cN2 − 2c4 vanishing only for ℓ = m; on the contrary, the func-
(cid:18) (cid:19) (cid:18) (cid:19)
tions F (a,z,ω) and G (a,z,ω) have nonvanishing
1 ℓm ℓm
Gℓm(a,z,ω)= nmqℓ−nℓqm N5/2 non-diagonal terms, yielding non-diagonal contributions
(cid:18) (cid:19) to the field susceptivity.
a2z2 ω2z2 a2z2
1+ + 1+ . (25)
× c4 c2 4c4
(cid:18) (cid:18) (cid:19)(cid:19)
7
The symmetric correlation function for the atoms is the atomic excited and ground states.
1
CA/B(τ,τ′)= ψ rA(τ),rB(τ′) ψ
ℓm 2h ±| ℓ m | ±i
Theresonanceinteractionbetweenthetwoaccelerated
=±21 eiω0(τ−τ′)+e−(cid:8)iω0(τ−τ′) (rgAe(cid:9))ℓ(reBg)m,(26) atoms is now obtained substituting Eqs. (21) and (26)
(cid:16) (cid:17) into Eq. (18), and taking the limits τ0 , τ .
→ −∞ → ∞
where subscripts eg indicate a matrix element between After simple algebraic manipulations, we obtain
δE = (µA) (µB) V (a,z,ω ) (µA) (µB) W (a,z,ω ), (27)
± eg ℓ ge m ℓm 0 ± eg ℓ ge m ℓm 0
where the explicit expressions of V (a,z,ω ) and expected. For example, in the case of atomic dipoles
ℓm 0
W (a,z,ω ) are with the same direction alongone ofthe axies x,y,z, for
ℓm 0
za/c2 1weobtainfrom(27)[inthiscaseonlytheterm
≫
containing V (a,z,ω ) contributes],
1 ℓm 0
V (a,z,ω )= f (a,z,ω )sin(ω S)
ℓm 0 z3 ℓm 0 0
(cid:20)
1
−gℓm(a,z,ω0)cos(ω0S)(cid:21), (28) δE ≃±(µAeg)ℓ(µBge)mz3((δℓm−qℓqm−2nℓnm)
2ω z 2ω c az ω2z2 2c2
a 0 sin 0 ln 0
W (a,z,ω )= F (a,z,ω )sin(ω S) × c a c2 − c2 za
ℓm 0 2z2c2 ℓm 0 0 (cid:20) (cid:16) (cid:18) (cid:19)(cid:17) (cid:18) (cid:19)
(cid:20)
2ω c az 8c2
0
−Gℓm(a,z,ω0)cos(ω0S) . (29) ×cos a ln c2 +qℓqm az
(cid:21) (cid:16) (cid:16) (cid:17)(cid:17)(cid:21) (cid:18) (cid:19)
These quantities explicitly depend on the atomic ac- 2ω c az
0
cos ln . (32)
cinetleerraatcitoionnapnodteanrteiatlhteo tgheenecraasleizoaftiaocnceolfertahteedstaattoimonsa.ry × (cid:16) a (cid:18)c2(cid:19)(cid:17))
The expression given above is valid for any value of
Our result (32) shows that the far-zone resonance in-
az/c2. Asinthescalarfieldcase,wenowinvestigatetwo
teraction between two accelerated atoms decreases with
limitingcases,z c2/aandz c2/a. Itiseasytoshow the distance as z−2 if the two dipoles are along z or y or
≪ ≫
that,forza/c2 1,thelinearsusceptibility(21)isfairly as z−4 it they are along x, compared with the z−1 de-
≪
well described by its stationary counterpart. Therefore,
pendence for atoms at rest. Also, a comparisonwith the
atthelowestorderinza/c2,werecovertheknowninertial
case of the scalar field discussed in the previous section,
resonance interaction [25]
showsthe emergenceofanotherchangeoftheresonance-
δE = (µA) (µB) V (ω ,z), (30) interactionbehaviororiginatingfromthestructureofthe
± eg ℓ ge m ℓm 0 field susceptibility, ultimately related to the vector na-
where V (ω ,z) is the well-known tensor potential ture of the electromagnetic field and to the spatially ex-
ℓm 0
tended character of our two-atom system (and not only
1 zω tothepresenceofthemetricfactorN(z,a)). Infact,Eq.
0
V = (δ 3n n ) cos(ω z/c)+
ℓm z3( ℓm− ℓ m (cid:20) 0 c (32) clearly shows that it is possible to control the ef-
fects of atomic acceleration on the resonance interaction
z2ω2 by an appropriate choice of the orientation of the two
sin(ω z/c) (δ n n ) 0 cos(ω z/c)(3,1)
× 0 (cid:21)− ℓm− ℓ m c2 0 ) dipole moments; for example, the resonance interaction
is stronglysuppressedif the two dipole moments areori-
In particular, Eq. (30) yields a potential energy as ented along the x direction, which is along the direction
z−1inthefar-zoneR λ,andas z−3inthenear-zon∼e ofacceleration. Also,whenthe dipoles areorthogonalto
≫ ∼
R λ, with λ being the atomic transition wavelength eachother,withoneofthemalongz andtheotheronein
≪
[25]. the(x,y)plane,thetermin(27)containingV (a,z,ω )
ℓm 0
On the other hand, at higher orders in az/c2, the cor- vanishesandonlythatwithW (a,z,ω )survives. Such
ℓm 0
rectionsduetotheatomicacceleratedmotiongiveaqual- a non-diagonal term is present only for a = 0, and thus
6
itative change of the distance dependence of the reso- its contribution is a unique signature of the accelerated
nance interaction, scaling with a different power law, as motionyielding, in this specific configuration(where the
8
interaction for stationary atoms is zero), a nonvanish- cial support from China Scholarship Council (CSC)
ing interaction energy. These properties suggest further
possibilities to single out the effect of the acceleration
by an appropriate choice of the orientation of the dipole Appendix A: Vacuum fluctuations and radiation
moments of the two accelerated atoms, which are not reaction: the general formalism
present for a single accelerated atom in the unbounded
space. In this Appendix, we briefly recall the procedure of
Dalibard et al [37], that we have used in this paper to
separate the contributions of vacuum fluctuations and
IV. CONCLUSIONS AND FUTURE radiation reaction to the resonance interaction between
PERSPECTIVES.
the two atoms. We assume a linear coupling between
atoms and field, as in Eq. (1).
In this paper we have investigated the resonance in- The solution of Heisenberg equations for the field op-
teraction between two uniformly accelerated atoms, one erators can be separated in a free part (af), which is
k
excited and the other in the ground state, prepared in a present even in the absence of interaction, and a source
correlated (symmetrical or anti-symmetrical) state, and part (as), which is related to the interaction between
k
interacting with the massless scalar field or the electro- atoms and field,
magneticfieldintheirvacuumstate. Wehaveconsidered
the contributions of vacuum fluctuations and of the ra- ak(t(τ))=afk(t(τ))+ask(t(τ)) (A.1)
diation reaction field to the resonance interaction, and
shownthat the Unruh thermalfluctuations do not affect with
the interatomicinteraction,whichisexclusivelygivenby
the radiation reaction term. We have shown that non- afk(t(τ))=ak(t(τ0))e−iωk(t(τ)−t(τ0)) (A.2)
thermal effects appear and that the acceleration yields
a change of the distance dependence of the resonance and
interaction. Specifically, we have shown that these non-
thermal effects, related to the non-inertial character of iλ τ
acceleration,resultinadifferentscalingwiththedistance ask(t(τ))= ~ dτ′σ2,A φ(xA(τ′)),ak(t(τ′))
and a different dependence on the acceleration as ex- (Zτ0 (cid:2) i
τ
peOctuedrrfersoumltsthoepeknnotwhenwUanyrfuohr-nteemwpfeurtautruerdeeevqeuloivpamleennctes., ×e−iωk(t(τ)−t(τ′))+ dτ′σ2,B φ(xB(τ′)),ak(t(τ′))
for example the impact of the change of interatomic po- Zτ0 (cid:2) i
tential on the Dicke phase transition or on macroscopic e−iωk(t(τ)−t(τ′)) (A.3)
phenomena such as thermodynamical properties of sys- × )
tems composed of many accelerating particles. These
points could be a promising direction to highlight non- (ananalogousseparationcanbeperformedfortheatomic
thermalsignaturesofrelativisticaccelerationsinamany- observables). Then, the equations of motion of a generic
body context. atomic observable (pertaining, for example, to the atom
A),O ,usingsymmetricorderingbetweenatomandfield
A
variables,canbe splittedinthe vacuum fluctuations and
ACKNOWLEDGMENTS self-reaction contributions
Financial support by the Julian Schwinger Founda- dOA dOA dOA
= + , (A.4)
tion,MIUR,UniversityofPalermo,isgratefullyacknowl- dτ dτ dτ
(cid:18) (cid:19) (cid:18) (cid:19)vf (cid:18) (cid:19)sr
edged. J. M. acknowledges support from the Alexander
Von Humboldt Foundation. W. Z. acknowledges finan- where
dO iλ
A = φf(x(τ))[σ (τ),O (τ)]+[σ (τ),O (τ)]φf(x(τ)) , (A.5)
dτ 2~ 2,A A 2,A A
(cid:18) (cid:19)vf (cid:16) (cid:17)
and
dO iλ
A = φs(x(τ))[σ (τ),O (τ)]+[σ (τ),O (τ)]φs(x(τ)) . (A.6)
dτ 2~ 2,A A 2,A A
(cid:18) (cid:19)sr (cid:16) (cid:17)
The procedure outlined above is exact. To obtain the vf and sr contributions to the time evolutionof the observ-
9
able O up to second-orderin the coupling constant, we and source parts, and substitute their expressions in the
A
nowseparatetheatomicoperatorsσ(τ)andO intofree equations above; after some algebra, we get
A
dO iλ
A = φf(x (τ))[σf (τ),Of(τ)]+[σf (τ),Of(τ)]φf(x (τ))
dτ 2~ A 2,A A 2,A A A
(cid:18) (cid:19)vf (cid:16) (cid:17)
λ2 τ
dτ′ φf(x (τ)),φf(x (τ′)) [σf (τ′),[σf (τ),Of(τ)]]+O(λ3), (A.7)
−2~2 A A 2,A 2,A A
Zτ0 (cid:8) o
and
dO λ2 τ
A = dτ′ φf(x (τ)),φf(x (τ′)) σf (τ′),[σf (τ),Of(τ)]
dτ −2~2 A A 2,A 2,A A
(cid:18) λ2 (cid:19)sτr Zτ0 (cid:2) in o
dτ′ φf(x (τ′)),φf(x (τ)) σf (τ′),[σf (τ),Of(τ)] +O(λ3). (A.8)
−2~2 B A 2,B 2,A A
Zτ0 (cid:2) in o
The average values of Eqs. (A.7) and (A.8) on the vac- effective Hamiltonians, H and H ,
vf sr
uum field state 0 , can be expressed in terms of the
| i
iλ2 τ
Heff = dτ′CF(x (τ),x (τ′)) σf (τ),σf (τ′) (A.9)
(cid:16) A (cid:17)vf −2~ Zτ0 A A h 2,A 2,A i
and
iλ2 τ
Heff = dτ′ χF(x (τ),x (τ′)) σf (τ),σf (τ′) +χF(x (τ),x (τ′)) σf (τ),σf (τ′) (A.,10)
(cid:16) A (cid:17)sr −2~ Zτ0 ( A A (cid:8) 2,A 2,A (cid:9) A B (cid:8) 2,A 2,B (cid:9))
wherewehaveintroducedthestatisticalfunctionsforthe Ananalogousseparationcanbeperformedfortheatomic
scalar field variables of atom B. The time evolution of the atomic
observables of atom A (B) can be now described by the
CF(x(τ),x(τ′))= 1 0 φf(x(τ)),φf(x(τ′)) 0(,A.11) effective Hamiltonian H +(Heff ) +(Heff ) .
2h |{ }| i A(B) A(B) vf A(B) sr
χF(x(τ),x(τ′))= 1 0 [φf(x(τ)),φf(x(τ′))] 0(A. .12)
2h | | i
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