The role of E1-E2 interplay in multiphonon Coulomb excitation

10 Pages·0.28 MB·English

by  Alexander Volya

The role of E1-E2 interplay in multiphonon Coulomb excitation
Alexander Volya and Henning Esbensen
Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
(Dated: February 8, 2008)
Inthisworkwestudytheproblemofachargedparticle,boundinaharmonic-oscillatorpotential,
being excited by the Coulomb field from a fast charged projectile. Based on a classical solution to
the problem and using the squeezed-state formalism we are able to treat exactly both dipole and
quadrupole Coulomb field components. Addressing various transition amplitudes and processes of
multiphonon excitation we study different aspects resulting from the interplay between E1 and E2
fields,rangingfrom classical dynamicpolarization effects toquestionsof quantuminterference. We
compare exact calculations with approximate methods. Results of this work and the formalism we
present can beuseful in studies of nuclear reaction physics and in atomic stopping theory.
3 PACSnumbers: 25.70.De,42.50.Dv
0
0
2 I. INTRODUCTION overallexcitation probability, the dipole and quadrupole
transition amplitudes, and their interplay are discussed.
n
a Recent studies of Coulomb-inducedbreakupof weakly We use a connection between classical and quantum
J boundnucleiindicateaconsiderablereductionofthedis- mechanics which is usually complex and in most cases

9 sociationprobabilityincomparisonwiththepredictionof takes an approximate quasiclassical form. However, in
1 thefirst-orderBorncalculation[1]. Thereductionisasso- situations where the dynamics of a system is deter-
ciated with dynamic polarization, where the quadrupole minedbylinearequationsofmotion,witharbitrarytime-
1
field creates a polarization that influences the effects in- dependent parameters, the classical-to-quantum corre-
v
7 duced by the dominating electric dipole. This is a well- spondence is exact. The harmonic oscillator which is
5 knownphenomenoninatomicstoppingtheory,wherethe drivenbythetime-dependenceofitsparameters,suchas
0 stoppingpowerforchargedparticlesdeviatesfromtheZ2 mass, frequency or external force, is an example of such
1 dependence ofBethe’s formula [2], Z being the chargeof a situation. Most theoretical developments in the field
0 a particle. Measurements, as for example, of the stop- ofparametricexcitationsofharmonicsystems weredone
3 ping power for protons and anti-protons[3], indicate the morethanthreedecadesago[17,18, 19,20, 21]. Despite
0
presenceofaZ3 correction. Overthe yearsasubstantial this, it is only during the past decade theorists began to
/
h progresshas beenmadeinmeasuringandunderstanding embracethebeautyofthisproblemandtodiscussitsrel-
-t of this phenomenon [4, 5, 6, 7, 8, 9, 10]. evance to many physical processes. Photon generation,
l lasers [22], quantum interference and two-photonexcita-
c In this work we consider the Coulomb excitation of
u a charged particle bound by a harmonic-oscillator po- tions [23], phase transitions,critical phenomena [24, 25],
n tential. The time-dependent electric dipole (E1) and informationtheory[26],iontraps,orientedchiralconden-
: sates[27]areexamplesofrecentdevelopments. Squeezed
v quadrupole (E2) fields of a projectile excite the parti-
states that appear as solutions in parametrically driven
i cle, and within this approximation the quantum prob-
X harmonic systems, also known as two-photon coherent
lem is treated exactly. This problem has been studied in
r thepast,startingfromtheclassicaltreatmentbyAshley, states,generalizetheconceptofcoherentstates[28]toin-
a
cludetwo-quantacouplingsintheradiationfield[29,30].
Ritchie, and Brandt [11], followed by quantum pertur-
bation theory [12], and later further perturbative Born Bypresentingthisworkwefurtherexpandapplications
terms [13, 14] have also been considered. Exact numer- of the squeezed state formalism to problems of Coulomb
ical solutions to the time-dependent quantum problem excitation. We focus our efforts on classical and quan-
havebeendemonstratedbyMikkelsenandFlyvbjerg[15]. tum effects resulting from the interplay of one-(dipole)
The use of a harmonic oscillator model, as was previ- and two-quanta (quadrupole) couplings of an oscillator
ously argued [13], is justified for reasons of simplicity system to an external field. Using symmetries we derive
and at the same time good quality of results in compar- simpleformulasforexcitationprobabilitiesandtransition
ison with observed stopping powers. The preset work amplitudes thatexplicitly showquantuminterferenceef-
addsyetanotherreason,asitdemonstrateshowclassical fects and can be used to improve perturbative calcula-
resultssuchas[11,16]canbe usedtoobtainexactquan- tions.
tum answers. Thus, the approach presented here does This work is divided into four sections. In Sec. II we
not involve heavy numerical calculations. It presents a review the properties of harmonic-oscillatorsystems and
simple and transparent way to understand and to treat the transition from a classical to a quantum description,
exactly the Coulomb excitation of an oscillator. As an whichisbasedoncanonicaltransformation. We summa-
application of the presented technique, rather than re- rize the properties of the parameters that describe the
peating previous numerical calculations, we concentrate initial state to final state transition (S-matrix) and de-
onalessexploredaspectofCoulombexcitation;herethe rivetransitionamplitudes andexcitationprobabilityrel-
2
evant to Coulomb excitation. Section III is dedicated to In order to simplify the notation we will write this in
applications. Here we consider the classical solution and matrix form as
address the transition to the quantum treatment. We
discuss quantum corrections, perturbative limits, charge b(t)=u(t)a+v(t)a†+c(t). (8)
asymmetry, and two-phonon excitations. The role of E1
The perturbation H drives the nontrivial time de-
and E2 interplay is emphasized in these discussions. We int
pendence of these creation and annihilation operators
summarize our findings in Sec. IV.
d
i b=[b(t), H ]= (t)+ −(t)b+ +(t)b†, (9)
int
dt F Q Q
II. PARAMETRIC EXCITATION OF COUPLED
OSCILLATOR SYSTEM where matrices
We consider a system of N oscillatorswith the unper- ±ll′(t)= Qll′(t) ei(ωl±ωl′)t (10)
turbed Hamiltonian Q 2√ωlωl′
p2 ω2q2 and a vector
H = l + l l . (1)
0
(cid:18) 2 2 (cid:19)
Xl (t)= Fl(t) eiωlt
l
Theseoscillatorsaredisturbedbyatime-dependentforce F √2ωl
Fl(t)andtime-dependentfrequencyparametersQll′(t)= areintroduced. The − and + arehermitianandsym-
Ql′l(t), expressed by metric matrices, respQectively.QThe time dependence of
1 the parameters u and v, as follows from (9), are deter-
Hint(t)= 2 Qll′(t)qlql′ + Fl(t)ql. (2) mined from
Xll′ Xl
d
ItisassumedthatHint(t)=0intheinfinitepastt i u= −u+ +v∗, (11a)
and in the infinite future t . First we quanti→ze−th∞is dt Q Q
→∞ d
problem in the infinite past. Here i v = −v+ +u∗, (11b)
dt Q Q
1
q = a e−iωlt+a†eiωlt , (3a)
l √2ωl (cid:16) l l (cid:17) and
ω d
p = i l a e−iωlt a†eiωlt , (3b) i c= −c+ +c∗+ . (12)
l − r 2 (cid:16) l − l (cid:17) dt Q Q F
wherea† anda aretime-independentcreationandanni- These equations are solved numerically with initial con-
l l
hilation operators for the l-th mode. For any other time ditions
we introduce creation and annihilation operators b†(t)
l u( )=1, v( )=0, c( )=0, (13)
and bl(t) so that the coordinates and momenta in the −∞ −∞ −∞
Heisenberg representation are which follow from (5). We stress here that solving Eqs.
1 (11)and(12)isequivalentto findinga fullclassicalsolu-
q = b (t)e−iωlt+b†(t)eiωlt , (4a)
l √2ωl (cid:16) l l (cid:17) tion to the problem. This follows from the fact that the
equations of motion are linear, so the time-dependence
ω
p = i l b (t)e−iωlt b†(t)eiωlt . (4b) of a Heisenberg operator becomes identical to the time-
l − r 2 (cid:16) l − l (cid:17) dependence of its classical analog.
The operators b†(t) and b(t) are defined in the interac-
tionrepresentation,as the time dependence due to H is
0
A. Properties of the S-matrix
introduced explicitly in Eq. (4). For the infinite past we
have by definition
In this work we will be interestedonly in the initial to
b( )=a, b†( )=a†. (5)
final state transition probabilities, i.e., in the S matrix
−∞ −∞
Att + b(t)andb†(t)alsobecometime-independent that comes from the transformation of operators
→ ∞
b=ua+va†+c, b† =u∗a†+v∗a+c∗. (14)
b( )=b, b†( )=b†. (6)
∞ ∞ The inverse transformationis
The transformation from the original operators a and
a†totheintermediatebandb†isdeterminedbyageneral a=u†b vTb†+c′, a† =uTb† v†+c′∗, (15)
− −
Bogoliubov transformation
where
bl(t)=Xl′ (cid:16)ull′(t)al′ +vll′(t)a†l′(cid:17)+cl(t). (7) c′ =−u†c+vTc∗. (16)
3
Transformation(14)iscanonicalandpreservescommuta- between the classical phase space distribution and the
tionrelationssothatif[al, a†l′]=δll′ then[bl, b†l′]=δll′; quantum density matrix can also be used to exactly re-
this puts some conditions on the matrices u and v construct the quantum solution from its classical analog
[35]. The application of the Wigner transformation in
uu† vv† =1, uvT vuT =0, (17a) studies of various properties of the harmonic oscillator
− −
uu† vTv∗ =1, u†v vTu∗ =0. (17b) has been demonstrated repeatedly [26, 36, 37].
− −
These linear canonicaltransformations,so-calledsym-
plectic transformations, are often encountered in classi- B. Survival probability
cal mechanics. The generators of the above symplectic
transformation form a group which generalizes the sim-
Here we assume that u, v and c are known. Using
ple SU(1,1) group [31, 32, 33] of a one-dimensional os-
these we determine transition amplitudes between quan-
cillator. In general any infinitesimal time evolution can
tum states. Coherent states,
be described by a linear canonical transformation which
preserves the phase space in accordance with Liouville’s
theorem. The beauty of our case is that the linearity re- α =e−|α|2/2 ∞ αn n , (18)
mains for any period of time evolution. This allows for | i nX=0√n!| i
an exact classical-to-quantum correspondence. Besides
themethodthatweimplementinthispaper,theWigner provide the best means for addressing this problem. In
transformation [34] that establishes the correspondence this basis the evolution operator U is [38]
1 1 1
β U α =A0exp (β2 + α2 )+ β†v(u∗)−1β∗ αT(u∗)−1vα+β†(u†)−1α β†(u†)−1c′ αT(u∗)−1c∗ .
h | | i (cid:18)−2 | | | | 2 − 2 − − (cid:19)
(19)
ThenormalizationconstantA0isdeterminedbythecom- The multidimensional Gaussian integral can be taken
pleteness condition as
πn
1 d2Nβ β U α 2 =1, (20) d2nbexp( bTAb+2sTb)= exp sTA−1s ,
πN Z |h | | i| Z − √detA (cid:0) (cid:1)
(24)
where integration goes through both real and imaginary where relevant to our case
parts of β. This is an important parameter, since it sets
β c 1 ρ ρ
the scale of all transitions and it determines the proba- b= ℜ , c= ℜ , A= −ℜ ℑ ,
bility to remain in the ground state P0,0 = 0U 0 2 = (cid:18)ℑβ (cid:19) (cid:18)ℑc (cid:19) (cid:18) ℑρ 1+ℜρ (cid:19)
|h | | i| (25)
A0 2. The quantity P is given by the Gaussian inte-
0,0
| |
gral, where α is set to zero for simplicity
1 ρ ρ
πN = d2Nβexp β 2+ (βTρβ+2cTβ 2c†ρβ) . s=QTc, Q=(cid:18) −−ℑℜρ −(1ℑ+ℜρ) (cid:19) . (26)
P Z −| | ℜ −
0,0 (cid:8) (cid:9) Becauseofthe previouslydiscussedpropertiesoftheBo-
(21)
goliubov transformation, inversion of the matrix A is
We introduce here the matrix
not needed. It can be shown that cTQA−1QTc =
ρ=v(u∗)−1, (22) c†c+ (cTρ∗c). This results in −
− ℜ
which is a characteristic of quadrupole excitation. Be- P = det(1 ρ†ρ) exp c†c+ (cTρ∗c) . (27)
00
sides some trivial normalization that will be further dis- q − − ℜ
(cid:2) (cid:3)
cussed,thematrixelementρ istheamplitudeofasingle The survival probability in the above form unifies the
ij
step two-quanta excitation of i-th and j-th modes. The rolesofE1andE2processes,andestablishesadirectand
classical time evolution of this matrix follows from the simple relation between the quantities of perturbation
time evolution of u and v (11), and is given by theory, ρ and c, and the exact answer. There is a clear
meaning of the different terms in the product (27). The
i dρ=ρ −∗+ −ρ+ρ +∗ρ+ +. (23) exp( c†c) represents the excitation of a coherent state
dt Q Q Q Q −
due to the dipole interaction; the term det(1 ρ†ρ)
−
This resultexplicitly showsthatρ isa symmetricmatrix is a result of quadrupole squeezing [27]. Tphe remaining
(ρ=ρT). part exp[ (cTρ∗c)] is a result of quantum interference.
ℜ
4
C. Transition Amplitudes Theenergytransfer,whichisimportantforcalculating
thestoppingpower,canbecalculatedusingEq. (14). For
the excitation from the ground state we obtain
Equation (19) can be used to obtain transition am-
plitudes. Coefficients in a Taylor expansion of (19) are
relatedtotransitionamplitudesbetweenoscillatorstates ∆E =h0| ωlb†lbl|0i= ωl|cl|2+ ωl|vll′|2. (33)
[27], with one-dimensional oscillator, for example, Xl Xl Xll′
β U α =e−(|α|2+|β|2)/2 nU m (β∗)nαm . (28) Terhgeyasbproevaedpcraensenatlssoabneacvaelrcaugleateenderagsyintr[a2n7s].fer. The en-
h | | i h | | i √n!m!
Xn,m
Belowwedenote by i,j,k... the normalizedstate con- D. One dimensional harmonic oscillator
| i
taining single quanta in i,j,k... oscillator modes (later
we will use directions x y and z). A single-quantum ex-
Thespecialcaseofaone-dimensionalharmonicoscilla-
citation amplitude is given by
torwithtime-dependentparametershasbeenextensively
studied previously [17, 18, 19, 40, 41, 42, 43]. According
j U 0 = A0 (u†)−1c′ =A0 (c ρc∗) , (29)
h | | i − { }j { − }j toEq. (27)theprobabilitytoremaininthegroundstate
is
where the subscript j denotes a component of the vector
in the brackets. Since the interaction vanishes in both P = 1 ρ2 exp c2[1 ρ cos(φ 2φ )] ,
00 ρ c
the infinite past and infinite future, the transition prob- −| | −| | −| | −
p (cid:8) (cid:9)(34)
abilities are symmetric with respect to initial and final
whereφ andφ arephasesofρandc,respectively. This
ρ c
states, in agreement with general principles of quantum
agrees with the previously obtained result [17]. We will
mechanics. However, the decay amplitude from an ex-
furtheronencounteraone-dimensionalcasewherec=0.
cited one-phonon state to the ground state, which is
Here only even-quanta transitions are possible, and the
excitation probabilities from the ground state and first
0U j = A0 (u∗)−1c∗ , (30)
h | | i − { }j excited state by 2n quanta are
differs from the excitation amplitude by a non trivial
(2n)!
phase. This emphasizes the effect of quantum interfer- P0,2n = 22n(n!)2 |ρ|2n 1−|ρ|2, (35)
encebetweendipoleandquadrupoleamplitudes. Setting p
either c or ρ to zero restores the symmetry.
The transitionamplitude fromthe groundstateto the
(2n+1)!
excited state with two oscillator quanta, requires expan- P = ρ2n(1 ρ2)3/2. (36)
sion of (19) to the order β2 1,2n+1 22n(n!)2 | | −| |
1+δ iU 0 j U 0
ij U 0 = ij A0ρ + h | | ih | | i . (31) III. COULOMB EXCITATION
h | | i p 2 (cid:18) ij A0 (cid:19)
We consider here the problem of Coulomb excitation
The first term in the above equation is due to a direct
ofaparticleboundinaharmonic-oscillatorpotential,by
quadrupoletransition,sinceitsamplitudeisdirectlypro-
a projectile of charge Z. We assume that the projec-
portional to ρ. The second term describes the second-
order dipole process. tile moves along a fixed trajectory R~(t) in the xy plane.
In order to utilize the treatment discussed above we ex-
For an initially excited system the transition ampli-
pandtheCoulombpotentialuptoquadratictermsinthe
tudes can be calculated in a similar manner, by expand-
particle coordinate q, namely, treating it in the dipole-
ing in both α and β; for example
quadrupole approximation
iU 0 0U j
iU j =A0(u†)−1+ h | | ih | | i. (32) C C
h | | i ij A0 V (t)= ~q R~ + 3(~q R~)2 q2R2 , (37)
coul R3 · 2R5 (cid:16) · − (cid:17)
Tailor expansion of a Gaussian form can be used to
where C = Ze2. The non-zero components of H de-
define Hermite polynomials, in particular for the multi- int
fined in Eq. (2) are
variable case such as Eq. (19). This, in turn, allows for
an analytic expression of transition amplitudes between
CR CR
oscillator states. Useful iterative relations between am- F (t)= x , F (t)= y , (38)
x R3 y R3
plitudesthatfollowfrompropertiesofHermitepolynomi-
alscanbeobtainedinthis way,oralternatively,iterative
relations can be derived directly from the properties of C C
the evolution operator [39]. Qxx = R5 3Rx2 −R2 Qyy = R5 3Ry2−R2 ,
(cid:0) (cid:1) (cid:0) (cid:1)
5
This leads to
3CR R C
Qxy =Qyx = Rx5 y , Qzz =−R3 . (39) c(x1) =−2iξxK1(ξx) , c(y1) =2ξyK0(ξy) , c(z1) =(04.4)
Theseexpressionsshowthatthemode inthe direction We note that c(1) is real, and c(1) is imaginary.
y x
perpendiculartothescatteringplaneisdecoupled,andit Using (23) the squeezing parameter ρ can be calcu-
is excited by the quadrupole term only. Further, we as- latedinasimilarBorn-typeperturbativemanner. Inthe
sume a straight line trajectory so that R~(t)=(b, vt, 0). lowest order we obtain
This does not simplify the problem, since our formal-
ism can be used for any trajectory. This assumption, ρxx = 4iqxxξx[K1(2ξx)+2ξK0(2ξx)] , (45a)
−
however, allows us to compare with previous results ρ =8iq ξ2K (2ξ ), (45b)
yy yy y 0 y
[11, 12, 13]. We introduce the following notations
ρ =4q ξ ξ (ξ +ξ )K (ξ +ξ ), (45c)
xy xy x y x y 1 x y
di = bv√C2ωi , qij = 2b2vC√ωiωj , (40) p ρzz =4iqzzξzK1(2ξz). (45d)
The above results can also be obtained using time de-
and
pendent perturbation theory. The first order excitation
ξi = bvωi , (41) pabroilbitaybitloityexincittehequi-atnhtadiri,ejctiionnaissiPnigIl=e s|tcei|p2.qTuahderupprooble-
process is to the lowest order PII = (1+δ )ρ 2/4.
where i and j are indices referring to x, y and z direc- i,j i,j | ij|
These results can be compared with the exact transition
tions. Often we will assume ω = ω = ω = ω; in that
x y z
amplitudes in Eqs. (29) and (31).
case the corresponding indices are omitted. The param-
In the sudden limit ξ 0 the dipole contribution
eters d and q can be viewed as dipole and quadrupole →
strengths, so that their ratio 1/(b√2ω) determines the comesfromthetransversedirection: cx ≈−2id,whilefor
quadrupole excitations ρ ρ 2q are dominant
applicability of the multipole expansion. | xx| ≈ | zz| ≈
in this limit. The adiabatic limit ξ is a textbook
→ ∞
exampleofpreservationofadiabaticinvarianceswithex-
A. Perturbative treatment ponential precision [45].
Here we will discuss several approximations. The first
B. Z3 corrections in classical solutions
one is the usual approximation relevant to multipole ex-
pansion of the Coulomb field. It utilizes the smallness
The long range dipole contributionis the mainpartof
of the photon wave length as compared to the size of
the cross section, thus any corrections to c are the most
the system. The next two limits are the “sudden limit”
important. We denote as c (t) the inhomogeneous part
whereξ 1andthe“adiabaticlimit”ξ 1.Finally,we 0
≪ ≫ of the solution to Eq. (12)
willusethetermBornapproximationwhichcorresponds
to some order in the iterative solution of the classical t
equations (11) and (12). Since the classical equations of c (t)= i (t′)dt′. (46)
0
− Z F
motionallowhereforanexactreconstructionofthewave −∞
function, the use of this terminology does not contradict
The c c ( ) = dc(1) is the previously discussed first
0 0
its quantum notion. ≡ ∞
Born dipole amplitude. The second Born correction,
All of the results in this study, such as in Eqs. (27),
which here simply means the second order approxima-
(29), and (31), are fully determined by the parameters
tion to the classical solution of Eq. (12) is
c and ρ, which may be referred to as shift (dipole) and
squeezing(quadrupole)parameters,respectively. Bothof ∞
iδc= [ −c (t)+ +c∗(t)]dt. (47)
these parameters can be expanded using the above ap- Z Q 0 Q 0
−∞
proximations. Each order of Born approximationresults
in the next order correction in terms of the quadrupole This integral can be studied numerically or perturba-
strength (40). Thus an expansion of c and ρ is a series tively with respect to ξ. In Fig. 1 the dependence of the
real and imaginary parts of c are shown as functions of
c=d c(n)(ξ)qn−1, ρ= r(n)(ξ)qn, (42) ξ.Intheadiabaticlimit,whichcorrespondstolargeξ all
n=X1,2,... n=X1,2,... of these functions decay exponentially: c(2) exp( 2ξ)
∼ −
as dictated by the adiabatic nature [45]. Approaching
where n=1,2... is the corresponding Born order. The
the sudden limit all corrections go to zero as powers of
coefficients c(n) and r(n) are ξ dependent only.
The lowestorder long-rangedipole approximationis a ξ.Intheξ 0limit c(x1) isthedominantterm,andthe
→ ℑ
textbooks example [44]. Using Eq. (12) we obtain lowest order correction to it is c(x2) 3πξ2. This leads
ℑ ≈
to the following approximationfor the probability [11]
∞ F (t)
c dc(1) = i l eiωltdt. (43)
≈ Z−∞ − √2ωl Px =4d2(1−3πqξ2). (48)
6
following approximate E2 contribution to the excitation
probability
2.5 1.0
0.8 8
2.0 2 ) 0.6 ∆Pρ =2q2(1+πqξ(1 4ξ))+2q2 1+ qξ . (52)
0.4 − (cid:18) 3 (cid:19)
1.5 (2) c/(3 00..02 The charge asymmetry in this case is of opposite sign,
1.0 -0.2
-0.4 resulting in an enhanced probability for repulsive inter-
2) 0.5 -0.60.0 0.5 1.0 1.5 action. This enhancement can only be observed in mea-
( c surements if the dominant dipole transitions are blocked
0.0
or restricted.
(2)
-0.5 Re(cx )
(2)
Im(cx )
-1.0 (2) C. Exact treatment
Re(cy )
(2)
-1.5 Im(cy )
We start this section by showing the comparison of
0 2 4 6 8 theexactexcitationprobabilityandthefirstBorndipole
approximation, P =c†c , see Fig. 2. The harmonic-
born 0 0
oscillator model is rather crude and can not be used to
fullyunderstandfeaturesoftherealisticCoulombdissoci-
ationofnuclei. However,itcanstillprovideaqualitative
picture. For example the Coulomb dissociation of 17F
FIG.1: Thelowestorderpolarizationcorrectionstothedipole 9
can be looked at assuming that the proton outside the
amplitudeareplottedasafunctionofξ.Theinsertshowsthe
behavior of ℑ(c(x2)) and ℜ(c(y2)) relative to 3πξ2. oxygen core is bound by a harmonic potential. The fre-
quency of the oscillator, related to the dipole excitation
energy, is assumed to be of the order of 1 MeV. In this
The above correction behaves as Z3 with the charge of caseaunitofthe oscillatorlengthcorrespondsto6.4fm.
the projectile, and reduces the excitation probability in Assuming Coulomb excitation by 56Ni we have C =6.3.
28
repulsive Coulomb scattering. This correction is com- The choice of parameters used in some figures below is
pletely classical as it appears in a solution of classical guidedbythismodel. InFig. 2weuseimpactparameter
differential equations, see [11]. Note that the homoge- b = 4 (b = 25.6 fm), C = 6, and the range of velocities
neouspartof(12)whichisthesourceofthispolarization from 0.2 to 20 (oscillator units) that corresponds to the
effect, is caused by the E2 field. incidentenergiesfrom0.02to 200MeV/u. The choiceof
Higher order corrections to the squeezing parameter impactparameterisrestrictedbytherangeofapplicabil-
ρ are relatively large but are generally less important ityofthe multipole expansion√2b 1.Ourdiscussions
≫
duetotheoverallsmallnessofthequadrupoleamplitude. alsoimply thatthe probabilityto findthe nucleoninthe
Hereagainweiterativelysolvetheclassicalequation(23). oscillator at a distance from origin larger than impact
This equation contains no dipole parameters, thus the parameter is negligible. For b=2 this probability is 4%,
correction on ρ is self-induced and would not change in assuming the ground state wave function. However for
the absence of the E1 field. Assuming that b=4 that we use, this is less than 10−3 %.
AsseeninFig. 2theeffectsofhigherordercorrections
t ∞
iρ (t)= +(t′)dt′ = dt′Θ(t t′) +(t′), (49) are significant. The a difference between repulsive and
0
Z−∞Q Z−∞ − Q attractiveinteractions is also transparent. This confirms
the recent results of more realistic calculations [1].
the second Born correctionis
∞
iδρ= dt ρ (t) −(t)+ −∗(t)ρ (t) . (50)
0 0
Z−∞ (cid:16) Q Q (cid:17) 1. Higher-order corrections
For the case of equal frequencies, this expression can be
Fromthe classicalsolutionsforρandcexactquantum
calculated analytically
results can be extracted using Eq. (27). In order to
δρ =8q2ξ[K (2ξ) 2ξK (2ξ)] iπq2ξ(1 2ξ)e−2ξ,(51a) discuss the impactof a quantumtreatment we canmake
xx 1 2
− − −
δρ = 2πq2ξ2e−2ξ 8iq2ξ2K (2ξ), (51b) anexpansionoftheexactanswerinEq. (27),whichgives
xy 1
− −
δρ = iπq2ξ(1+2ξ)e−2ξ, (51c)
yy
−
16 (c†c)2 1
δρzz =−8q2ξK1(2ξ)+ 3 iq2ξ2K1(2ξ). (51d) Px =1−P00 ≈c†c− 2 +2Tr(ρ†ρ)−ℜ(cTρ∗c) (53)
In the ξ 0 limit, excitations due to the quadrupole
→
field comes from the x and z directions, leading to the =P +∆P +∆P +∆P .
born c ρ inter
7

0.25
-1
10
0.20 First Born
C=6, b=4
n
C=-6, b=4 or
b
0.15 P
Px P/ 10-2
0.10
0.05 -3
10
0.00 0 5 10 15 20 25
0 5 10 15 20
v (oscillator units)
v (oscillator units)
FIG. 3: Corrections to the first Born E1 probability. Values
FIG. 2: Excitation probability as a function of incident ve-
arerelativetoP . Logarithmicscaleisusedtoshowcontri-
locity,at theimpact parameter b=4.Thedashedlineis the born
resultoftherepulsiveinteraction,C =6.Thedottedlinecor- butions of very different magnitude. Corrections ∆Pc/Pborn
responds to theattractive interaction C =−6.These results ∆Pρ/Pborn and ∆Pinter/Pborn are shown in dotted, dashed
andsolidlines,respectively. Parametersb=4andC =6 are
are compared to the first Born dipole approximation shown
used for thisfigure.
by thesolid line.
AlthoughitmayseemthatthisisaquantumBarkascor-
We identify here three maincorrectionsto the firstBorn
rection, correctionvanishes to lowest order. This follows
dipole excitation probability. The first is
from Eqs. (44) and (45).
∆P =c†c (c†c)2 P , InFig. 3theabsolutevaluesof∆Pc,∆Pρ,and∆Pinter
c − 2 − born relative to the first Born E1 probability are plotted as
functionsofvelocity. Thefiguredemonstratesthatdomi-
whichis a correctionto the dipole transitionprobability.
nantcorrectionstothefirstorderdipoleprobabilitycome
It is known that the excitation of a quantum oscillator
fromsecondandhigherorderdipoletransitions,andfrom
by an externalforce creates a coherentstate. The dipole
the lowest order quadrupole. The interference term con-
amplitude c, which is related to a classical shift of the
tributes only on the level of 1%, and quickly vanishes in
oscillator from its equilibrium position, appears in the
the sudden limit. As discussed above, the interference
exponent of Eq. (27). This accounts for all second and
term vanishes to order Z3. Thus it is a Z4 term to lead-
higher order E1 processes. Exponentiation of the dipole
ing order.
amplitude, inorder to accountfor the loss ofstrength to
multi-phonon excitations, has been previously discussed
[46]. We stress here that other parts of ∆P are of clas-
c 2. Z-dependence
sical origin and mainly Z3-dependent. They come from
the effect of the quadrupole field on the shift of the os-
In order to examine the Z-odd part of the excitation
cillator from its equilibrium. This creates a deviation of
probability from a different point of view we introduce
c from the Born result c as discussed in Sec. IIIB.
0 the Barkas factor
The next term in (53),
P (Z) P ( Z)
∆P =Tr(ρ†ρ)/2, B = x − x − . (54)
ρ
P (Z)+P ( Z)
x x
−
is a quadrupole contribution, which comes from the ex-
As discussed above this effect owes its existence to the
pansion of the square root in (27). Similar to the ex-
classical dynamic interplay between E1 and E2 fields.
ponent in the dipole term, the use of the square root
With the validity of a multipole expansion, the bulk
accounts for all higher order quadrupole transitions.
of this correction is expected to be proportional to the
Finally, a very interesting and purely quantum inter-
quadrupole amplitude q. In Fig. 4 B/q is plotted as a
ference effect appears through the last term in (53),
function of ξ. Various cases corresponding to different
∆P = (cTρ∗c). impactparametersandprojectilechargesareconsidered.
inter
−ℜ
8

5
2 0.0005
3 c=6, b=10
4 c=6, b=5 C=6, b=4
2 0.0004
>| C=-6, b=4
c=4, b=3 z
,
z
q 3 U|0.0003
B/ c=6, b=3 0|
- <
2 |0.0002
0.0001
1
0 0 5 10 15 20 25
0.0 0.5 1.0 1.5 2.0 2.5
v (oscillator units)
FIG. 5: The transition probability from the ground state to
FIG. 4: The Barkas factor as a function of ξ, for various pa-
rameters. The classical correction to the sudden limit, 3πξ2, the two-quantum excitated state in the z direction is shown
asafunctionofincidentvelocity. Solidanddottedlinescorre-
is shown by a dashed curve.
spondtorepulsiveandattractiveCoulombforce,respectively.
In the second Born approximation, the B/q is only a
function of ξ, thus independent of both charge and im- We introduce a quadrupole coupling strength x which
pact parameter [13]. Deviations observed on the plot simply scales the quadrupole part of the perturbation
emphasizethe significanceofothercorrectionsthatscale Hamiltonian (2). This can be viewed as scaling the
as higher powers in q and higher odd powers in Z. quadrupole strength q xq. The limits x = 0 and
→
The charge asymmetry related to the dipole- x = 1 naturally correspond to the cases when E2 is ne-
quadrupole interference appears naturally in various glected and when E2 is at its natural value, respectively.
transition amplitudes. The Fig. 5 shows the transi- Eq. (31), as one would expect from perturbation the-
tion probability from the ground state to the state with ory, shows that the amplitude for a two-phonon excita-
two quanta in z direction. In this transition the vanish- tion is given by the sum of two terms that reflect the
ing out-of-plane dipole force makes the quadrupole term firstorderquadrupoleandsecondorderdipoleprocesses.
dominant. As demonstrated by Eq. (52) the Z3 Barkas- The E1 and E2 interplay is not trivial here and comes
type charge dependence of the quadrupole term ρ is of in severalplaces. The first term in (31) at large quadru-
theoppositesignincomparisontothedipole-quadrupole ple strength is directly proportional to x. Thus in this
case shown in Figs. 2 and 4. In agreement with this, an limit a quadratic scaling of the excitation amplitude is
enhancement of the excitation probability in repulsive expected. At a smaller strength a substantial contribu-
Coulomb interaction is observed in Fig. 5. tion to the second term in (31) appears as a classical
dynamicpolarizationeffect,influencingthe amplitudec.
Thiscanreducetheexcitationprobabilityforarepulsive
interaction. Finally, the phases of terms in (31) can in-
3. Interplay of dipole and quadrupole fields
terfere. It must be noted that the dynamic polarization
disappears at very low and very high velocities. Also
Ourharmonic-oscillatormodelincorporatesE1andE2
both d 1/v and q 1/v but since dipole comes in
electromagneticprocessesexactly. Theroleandinterplay ∼ ∼
the second order, the quadrupole amplitude dominates
of these processes in multi-phonon excitation may be of
at high velocity leading to
interestinthephysicsofGiantDipoleResonances(GDR)
[46, 47]. Although precise calculations with inclusion of P (x)=4q2x2. (55)
0,2
the relevant microscopic physics of GDR are beyond the
scope of this paper, we consider here the effect that the InFig. 6thetwo-phononexcitationprobabilityasafunc-
E2processplaysinthetwophononexcitationprobability tion of the coupling strength x is shown relative to the
case with no E2, x = 0. Observed features are in agree-
P = ij U 0 2. ment with the above discussion. Cases corresponding to
0,2
|h | | i|
Xi<j repulsiveinteraction,shownbysolidlines,alwaysleadto
9
rameterscandρaredeterminedfromclassicalequations.

Our results are exact for a harmonic oscillator. For a
2.50 V=7, C=-6 generalcase they can be used as a parameterization,ex-
tending the higher-order (multi-phonon) dipole approxi-
2.25 V=7, C=6
) mation introduced by Norbury and Baur [46] to include
0
( V=3, C=-6 the quadrupole field and related interference effects.
0 2.00
P2, We discussed the interplay between the electric dipole
)/ 1.75 and quadrupole fields and the resulting effects on vari-
(x0 V=5, C=6 ous excitation probabilities. In the intermediate range
P2, 1.50 of energies, between instant and adiabatic limits, (0.5 <
ξ < 7) the influence of dynamic polarization dominates.
1.25
It was shown that the origin of this effect is purely clas-
V=3, C=6 sical. The quadrupole polarization influences the effect
1.00
of the dipole force, leading to a significant change in the
excitationprobability. Thisphenomenon,knowninstop-
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ping power theory as Barkas effect, in the lowest order
x leadstoaZ3 chargedependence,andreducestheexcita-
tion probability for repulsive Coulomb interactions. We
foundasimilarclassicaleffectofself-inducedpolarization
in the quadrupole amplitude. This effect is of the oppo-
site signand for the same repulsive kinematics results in
FIG.6: Relativetwo-phononexcitationprobabilityasafunc-
anenhancementofthequadrupoleamplitude. Theeffect
tion of the quadrupolestrength.
wasdemonstratedinz-polarizedtwo-phononexcitations.
Our exact results exhibit some additional higher-order
quantum contributions, but corrections from these are
reducedprobabilityascomparedtokinematicallyidenti-
small. We presented a number of numerical calculations
cal situations but attractive interaction (dashed lines).
and plots that demonstrate a variety of observable phe-
The dynamic polarization effect is maximized around
nomena that can be attributed to the E1-E2 interplay.
v =3whichisconsistentwiththeenhancedchargeasym-
metry observed in Fig. 2. In this velocity regime, the By presenting this work we hope to introduce the co-
polarization effect is so strong that for repulsive interac- herent plus squeezed-state formalism to the field of nu-
tions “turning on” the quadrupole field (x =1) leads to clearreactionphysicsandnuclearexcitations. Numerous
a reduction of a two-phonon excitation probability. developmentsthatusedipoleexcitationscanbeextended
toincludequadrupoletransitions. Noteveryphysicalsit-
uation can be described by a harmonic oscillator. How-
IV. CONCLUSIONS ever,armedwithanexactsolutionandafullsetoftime-
dependent wave functions, perturbation theory basedon
squeezed states can be a promising future direction.
Inthisworkweconsideredasimplemodelofacharged
particle in a harmonic-oscillator potential. The system
is excited by the electric dipole and quadrupole fields
of a charged projectile passing by. We have devel-
Acknowledgments
oped an application of the squeezed-state formalism to
solve this problem exactly. In terms of the one-phonon
dipoleexcitationamplitudescandthedirecttwo-phonon The authors are thankful to Carlos A. Bertulani for
quadrupole excitation amplitudes ρ we derived simple usefuldiscussions. ThisworkwassupportedbytheU.S.
expressions for various transition amplitudes, and the Department of Energy, Nuclear Physics Division, under
totaland multi-phononexcitationprobabilities. The pa- contract No. W-31-109-ENG-38.
[1] H. Esbensen and G. F. Bertsch, Nucl. Phys. A706, 477 Methods in Physics Research 204, 605 (1983).
(2002). [6] L. Porter and H. Lin, Journal of Applied Physics 67,
[2] H.Bethe, Ann.d. Phys.5, 325 (1930). 6613 (1990).
[3] L. Andersen, P. Hvelplund, H. Knudsen, S. Moller, [7] D. Novkovic, K. Subotic, M. Stojanovic, Z. Milosevic,
J.Pedersen,E.Uggerhoj, K.Elsener,andE.Morenzoni, S.Manic,andD.Paligoric, JournaloftheMoscowPhys-
Phys.Rev.Lett. 62, 1731 (1989). ical Society 3, 209 (1993).
[4] H.Andersen,Physica Scripta 28, 268 (1983). [8] J. Pitarke, R. Ritchie, P. Echenique, and E. Zaremba,
[5] L. Porter and R. Jeppesen, Nuclear Instruments and Europhysics Lett. 24, 613 (1993).
10
[9] N.AristaandA.Lifschitz,Phys.Rev.A59,2719(1999). [27] A. Volya, S. Pratt, and V. Zelevinsky, Nucl. Phys. A
[10] P.Leung, Phys. Rev.A 40, 5417 (1989). A671, 617 (2000).
[11] J. Ashley, R. Ritchie, and W. Brandt, Phys. Rev. B 5, [28] R. Glauber, Phys. Rev. 131, 2766 (1963).
2393 (1972). [29] H. Yuen,Phys.Rev.A 13, 2226 (1976).
[12] K.Hill and E. Merzbacher, Phys.Rev.A 9, 156 (1974). [30] J. Hartley and J. Ray,Phys.Rev. D 25, 382 (1982).
[13] H. Mikkelsen and P. Sigmund, Phys. Rev. A 40, 101 [31] G.ProfiloandG.Soliana,Phys.Rev.A44,2057(1991).
(1989). [32] C. Gerry,Phys. Rev.A 35, 2146 (1987).
[14] H. Mikkelsen and H. Flyvbjerg, Phys. Rev. A 42, 3962 [33] W. Lianfu, W. Shunjin, and J. Quanlin, Zeitschrift fur
(1990). Physik B (Condensed Matter) 102, 541 (1997).
[15] H. Mikkelsen and H. Flyvbjerg, Phys. Rev. A 45, 3025 [34] E. Wigner, Phys. Rev. 40, 749 (1932).
(1992). [35] H. Esbensen, in International School of Physics, Enrico
[16] J.D.JacksonandR.L.McCarthy,Phys.Rev.B6,4131 Fermi, on Nuclear Structure and Heavy Ion Reactions,
(1972). edited byR.Broglia, C. Dasso, and R.Ricci (NuovoCi-
[17] A. Baz, I. Zeldovich, and A. Perelomov, Scattering, re- mento, 1981), p.571.
actions and decay in nonrelativistic quantum mechanics. [36] G. Agarwal and S. Arun Kumar, Phys. Rev. Lett. 67,
(Rasseyanie, reaktsii i raspady v nerelyativistskoi kvan- 3665 (1991).
tovoi mekhanike) (Jerusalem, Israel Program for Scien- [37] D. Han, Y. Kim, and M. Noz, Phys. Rev. A 37, 807
tificTranslations, 1969). (1988).
[18] V.PopovandA.Perelomov,JETPLett.29,738(1969). [38] B. R.Mollow, Phys. Rev. 162, 1256 (1967).
[19] V.PopovandA.Perelomov,JETPLett.30,910(1970). [39] F. Fernandez and R. Tipping, J. Chem. Phys. 91, 5505
[20] H.Lewis, Phys. Rev.Lett. 18, 510, 636 (1967). (1989).
[21] H. Lewis and W. Riesenfeld, J. Math. Phys. 10, 1458 [40] M. Sebawe Abdalla, Phys. Rev.A 34, 4598 (1986).
(1969). [41] M. Sebawe Abdalla, Phys. Rev.A 33, 2870 (1986).
[22] M. Abdalla, M. Ahmed,and S. Al-Homidan, J. Phys. A [42] S. Abeand R.Ehrhardt,Phys. Rev.A 48, 986 (1993).
31, 3117 (1998). [43] K. Husimi, Prog. Theor. Phys. 9, 381 (1953).
[23] N. Georgiades, E. Polik, and K. H.J., Phys. Rev. A 59, [44] C. Bertulani and G. Baur, Phys. Rep. 163, 299 (1988).
676 (1999). [45] L.LandauandE.Lifshitz,Mechanics, Courseoftheoret-
[24] A.Isar, Fortschritteder Physik 47, 855 (1999). ical physics vol.1 (Oxford;NewYork: Pergamon Press,
[25] P. Drummond,S. Chaturvedi, K. Dechoum, and J. Cor- 1976).
ney (2001), vol. 56A, p. 133. [46] J.W.NorburyandG.Baur,Phys.Rev.Cp.1915(1993).
[26] J. Aliaga, G. Crespo, and A. N. Proto, Phys. Rev. Lett. [47] C. Bertulani, L. Canto, M. Hussein, and
70, 434 (1993). A. deToledo Piza, Phys. Rev.C 53, 334 (1996).