Analytical solution to the Schrodinger equation of a laser-driven correlated two-particle system

5 Pages·0.11 MB·English

by  Uwe Schwengelbeck

Analytical solution to the Schr¨odinger equation of
a laser-driven correlated two-particle system
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0
Uwe Schwengelbeck
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2 Departamento de F´ısica Aplicada, Universidad de Salamanca,

n E-37008 Salamanca, Spain
a
J February 1, 2008

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3

Abstract
1
v Thetime-dependentquantumsystemoftwolaser-drivenelectronsina
8 harmonicoscillatorpotential,isanalysed,takingintoaccounttherepulsive
4 Coulomb interaction between both particles. The Schr¨odinger equation
1 of the two-particle system is shown to be analytically soluble in case of
1 arbitrary laser frequencies and individual oscillator frequencies, defining
0 the system. Quantum information processing could be a possible field of
2 application.
0
/ PACS numbers: 42.50.Ge, 03.65.Ge, 33.80†
h
p There is an inherent interestin analyticaland non-perturbative solutions of
- time-dependent quantum problems, all the more considering quantum systems
t
n withmorethanoneparticle. Oneofthefewexamplesofsuchkindisthesystem
a of two electrons in an electromagnetic field [1], where an analytical solution of
u
the Schr¨odinger equation has been given recently [2]. The aim of this work is
q
to demonstrate that exact wavefunctions are available for the time-dependent
:
v
problem of two correlated charged particles in a harmonic oscillator potential,
i
X subject to a laser field (see also Ref. [3]). The given analytical solutions [3] are
r valid for arbitrary driving laser frequencies and individual oscillator frequen-
a cies, defining the system. A possible field of application might be quantum
information processing (cf., e.g., Ref. [4]). Another application of the time-
dependent solution to the two-particle Schr¨odinger equation could be the test
of costly numerical algorithms to simulate the dynamics of helium in intense
laserfields—still oneofthe topicalproblemsintheoreticalatomic physicsthese
days. The latter system has, apart from the Coulomb interaction between the
electrons and the nucleus, a similar structure as the system considered here,
containing a harmonic potential.
ThewavefunctionΨoftwolaser-drivenparticlesofequalmassµandcharge
q in a harmonic potential, associatedwith the oscillatorfrequency Ω, obeys the
Schr¨odinger equation
∂
i¯h Ψ(r ,r ,t)=Hˆ(r ,r ,t)Ψ(r ,r ,t), (1)
1 2 1 2 1 2
∂t
1
where the Hamiltonian in length gauge [5] reads
2 pˆ2 µ
Hˆ(r ,r ,t)= i + Ω2r 2−qr ·eˆE sin(ωt+δ) +V(r ,r ), (2)
1 2 i i 0 1 2
2µ 2
i=1(cid:20) (cid:21)
X
ω and E are the frequency and the amplitude of a monochromatic laser field,
0
respectively, eˆ denotes a unit polarization vector, and
q2
V(r ,r )= (3)
1 2 |r −r |
1 2
describes the represents Coulomb interaction between both particles.
On using center of mass and relative coordinates,
1
R= (r +r ) and r=r −r , (4)
1 2 1 2
2
the Hamiltonian (2) decouples,
Pˆ2 pˆ2 µ q2
Hˆ = + +µΩ2R2+ Ω2r2−2qR·eˆE sin(ωt+δ)+ , (5)
4µ µ 4 0 |r|
where
1
Pˆ =pˆ1+pˆ2 =−i¯h∇R, and pˆ = (pˆ1−pˆ2)=−i¯h∇r, (6)
2
and the Schr¨odinger equation (1) admits solutions of the factorized form
Ψ=ψ(R,t)φ(r,t)χ(1,2), (7)
where χ(1,2) is the corresponding spin state of the two-particle system. The
respective differential equations for the wavefunctions ψ(R,t) and φ(r,t) =
φ(r)exp(−iǫt/¯h) in (7) then read
∂ Pˆ2
i¯h ψ(R,t)= +µΩ2R2−2qR·eˆE sin(ωt+δ) ψ(R,t) (8)
0
∂t 4µ
" #
and
pˆ2 µ q2
ǫφ(r)= + Ω2r2+ φ(r). (9)
µ 4 |r|
(cid:18) (cid:19)
For both of the above equations, (8) and (9), analytical solutions are available
[3].
Equation (8), associated with the center of mass motion of the two parti-
cles, has essentially the form of the Schr¨odinger equation of a driven harmonic
oscillator. Given an initial wavefunction ψ(R,0), the solution of equation (8)
can be obtained by means of a path integral [6]:
∞
ψ(R,t)= K(R,t;R′,0)ψ(R′,0)dR′, (10)
−Z∞
2
where the propagator,corresponding to equation (8), reads
K(R,t;R′,0)= µΩ eh¯iS(R,t;R′,0), (11)
siπ¯hsin(Ωt)
containing the action function
µΩ
S = (R′2+R2)cosΩt − 2R′·R
sin(Ωt)
2qE(cid:2) t
+ 0 R·eˆ sin(ωτ +δ)sinΩτdτ
µΩ
Z0
2qE t
+ 0 R′·eˆ sin(ωτ +δ)sin[Ω(t−τ)]dτ
µΩ
Z0
2q2E2 t τ
− 0 sin(ωτ +δ)sin(ωs+δ)sin[Ω(t−τ)] sinΩsdsdτ .(12)
µ2Ω2
Z0 Z0 (cid:21)
As anexample weshallconsiderherealaserfieldwith linearpolarizationalong
the z-direction and phase δ = 0. On starting with the unperturbed oscillator
ground state
3/4
ψ(R,0)= 2µΩ e−hµ¯ΩR2, (13)
π¯h
(cid:18) (cid:19)
the center of mass wavefunction at time t, given by integral (10), reads (R =
(X,Y,Z))
ψ(R,t) = 2µΩ 3/4e−32iΩte−hµ¯Ω(X2+Y2)eihµ¯ΩZ2cotΩt
π¯h
(cid:18) (cid:19)
µΩ qE sinωt−ΩωsinΩt −Z 2
0 µ(Ω2−ω2)
×exp −
 (cid:16)¯hsin2Ωt (1−icotΩt) (cid:17) 
 
 ωcosωt − Ωsinωt cotΩt
×exp 2iqE Z
0 ¯h(Ω2−ω2)
(cid:18) (cid:19)
sinωt cosωt ω+ Ω2 −2ΩcotΩt sin2ωt
×exp −iq2E2 ω
 0 (cid:16)2µ¯h(Ω2(cid:17)−ω2)2 
 iq2E2t 
×exp 0 , (14)
2µ¯h(Ω2−ω2)
(cid:18) (cid:19)
whichcanbeverifieddirectlybysubstitutionintoequation(8). Itmaybenoted
that for the case of a circularly polarized laser field, the wavefunction can be
obtained in a similar way as in the above case with a linearly polarized laser
field.
Theremaningproblem(9),concerningtherelativemotionofbothelectrons,
is analytically soluble for an infinite denumerable set of oscillator frequencies
3
Ω [7, 8, 9]. On using spherical polar coordinates, the relative coordinate wave-
function reads
u(r)
φ(r)= Y (θ,φ), (15)
lm
r
whereY arethe sphericalharmonics,andthe radialpart,givenby the ansatz
lm
∞
u(r)=rl+1e−4µh¯Ωr2 aνrν (16)
ν=0
X
is determined by
¯h2 d2 µ q2 ¯h2l(l+1)
ǫu(r)= − + Ω2r2+ + u(r). (17)
µ dr2 4 r µr2
(cid:20) (cid:21)
In closed form solutions of the radial equation (17), only a finite number of
coefficients a 6= 0 contributes to the power series expansion in equation (16),
ν
whichis satisfiedinparticularcasesofharmonicoscillatorfrequenciesΩ. Asan
example, an account of closed form solutions shall be given here for oscillator
frequencies,relatedtotheangularmomentumquantumnumbersl =0,1,...,∞
by (cf. Ref. [8])
q4µ
Ω= . (18)
2h¯3(l+1)
The eigenfunction (15)for the relativemotion ofthe two particlesis then given
by
φ(r)=rle−4µh¯Ωr2 1+ q¯h2Ωr Ylm(θ,φ), (19)
(cid:18) (cid:19)
corresponding to an energy ǫ = h¯(3Ω + q4µ). For a numerical treatment of
2 h¯3
equation (9) yielding solutions valid for arbitrary values of oscillator frequency
Ω, the reader may be refered to Refs. [10, 11].
It is noted that the center of mass wavefunction (14), comprising the dy-
namical evolution of the system subject to the laser field, remains symmet-
ric against particle exchange. At the same time, the total wavefunction Ψ =
ψ(R,t)φ(r,t)χ(1,2) has, according to the Pauli principle, to be antisymmetric
againstparticle exchange. Thus, in case of antisymmetric solutions of equation
(9) with φ(r) = −φ(−r), the spin state χ(1,2) of the system has to be sym-
metric against particle exchange, leading to triplet states. On the other hand,
symmetricfunctions φ(x) mustbe accompaniedbyanantisymmetricspinstate
χ(1,2)=−χ(2,1),describingsingletstatesofthe system. Thelaserinteraction
does not change the parity of the system wavefunction. In case of a vanishing
harmonicoscillatorpotential, i.e.Ω=0 in Hamiltonian(2), the systemreduces
to that oftwo “free” particles in a laser field, discussedin Ref. [2]. The relative
motion of both particles is then represented by repulsive Coulomb wavefunc-
tions,satisfyingeitheringoingoroutgoingboundaryconditions. Thesesolutions
do not permit a motion of both particles with equal kinetic energy and wave
vectors in the same direction [2]. In contrast to that case, the time-dependent
4
center of mass wavefunction (8) and the stationary relative coordinate wave-
function (9) of the system, based on the repulsive Coulomb and the harmonic
potential term, describe two laser-driven particles of which both contributions
are always pointed towards the same direction.
Inconclusion,thetime-dependentproblemoftwolaser-drivencorrelatedpar-
ticles in a harmonic potential has been shown to be analytically soluble. The
Hamiltonian of the Schr¨odinger equation of the system can be decoupled into
that ofa drivenharmonic oscillatorfor the center ofmass motion anda Hamil-
tonian for the relative motion, comprising the repulsive Coulomb interaction
between both particles. For both the center of mass part and the relative part
of the two-particle problem, analytical solutions are available. Besides applica-
tions concerning laser interaction with ions in a trap, and nuclear systems, the
time-dependent wavefunctions may be utilized to check numerical algorithms
to determine the dynamical evolution of the wavefunction of two-electron sys-
tems, e.g. to simulate the dynamics of the helium atom in laser fields. Another
possible field of application might be quantum information processing.
Acknowledgments
It gives me great pleasure to thank Luis Plaja, Luis Roso, Javier Rodr´ıguez
Va´zquezdeAldanaandRobertoNumicoforthestimulatingexchangeanduseful
discussions.
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