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Revista Mexicana de Física
ISSN: 0035-001X
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Sociedad Mexicana de Física A.C.
México
Laroze, D.; Rivera, R.
On the transition probability for a quantum dot in a time dependent magnetic field
Revista Mexicana de Física, vol. 53, núm. 7, diciembre, 2007, pp. 112-116
Sociedad Mexicana de Física A.C.
Distrito Federal, México
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REVISTA MEXICANA DE FÍSICA S 53 (7) 112–116
DICIEMBRE 2007
On the transition probability for a quantum dot in a
time dependent magnetic field
D. Laroze∗ and R. Rivera
Instituto de Fı́sica, Pontificia Universidad Católica de Valparaı́so,
Casilla 4059, Valparaı́so, Chile
Recibido el 30 de noviembre de 2006; aceptado el 8 de octubre de 2007
In this work we analyze in detail the dynamical behavior of a quantum dot in presence of a uniform time-dependent magnetic field, in the
effective mass approximation. An exact solution for the time-evolved wave function is obtained when the initial state is a particular Fock–
Darwin state. Moreover, we calculate the transition probability from an initial Fock-Darwin state to a given final Fock-Darwin state for the
case of a magnetic field that changes linearly in time, flipping its direction.
Keywords: Quantum Dot; Exact Solution; Time dependent Problems.
En este trabajo analizamos en detalle, en la aproximación de masa efectiva, la dinámica de un electrón en un punto cuántico en presencia
de un campo magnético uniforme dependiente del tiempo. Se encuentra una solución exacta para la evolución temporal de la función de
onda cuando el estado inicial es un estado de Fock-Darwin. Se calcula también la probabilidad de transición entre estados de Fock-Darwin
especı́ficos cuando el campo magnético invierte su dirección cambiando linealmente en el tiempo.
Descriptores: Cuánticos; soluciones exactas; problemas dependientes del tiempo.
PACS: 73.63.Kv, 73.21.La, 73.21.–b
1. Introduction
Quantum dots (QD) are zero dimensional objects constructed
by patterning and epitaxial growth techniques in semiconductor heterostructures [1]. An important characteristic of these
systems is that the phase coherent length of the electron wave
functions exceeds the size of the dots, and consequently, their
motion is considered ballistic. There exists a vast amount
of literature discussing diverse physical aspects of these systems, like spectroscopy, nonlinear optics, magnetotransport,
many-body effects, etc [2]. In many theoretical studies of
the QD, the electrons are considered independent, and their
energy level structure is determined accordingly. In addition, it is usually assumed that the lateral con?ning potential
has a parabolic shape. Apart from the obvious mathematical
simplicity brought about by this assumption, self-consistent
calculations, as well as experimental observations, have provided a support to the validity of this approach [3].
On the other hand, the dynamics of electrons in magnetic
fields has played a fundamental role in physics and its application to technology in a wide spectrum of topics, such as the
Aharonov – Bohm effect [4–7], the bidimensional electron
gas at the interface of semiconductor heterostructures [8], and
electromagnetic lenses with time-dependent magnetic fields
[9–12]. All these situations are interesting by themselves,
and present rather complex mathematical structures; therefore, exact analytic solutions have been found only in a few
special cases [13]. Therefore, the study of electrons in a QD
under the presence of the magnetic field is important and in
order.
In the present work, we study the dynamics of electrons of a quantum dot interacting with a homogeneous timedependent magnetic field together with the corresponding in-
duced electric field, assuming a parabolic dot con?ning potential in the effective mass approximation. We find an exact
solution for the corresponding propagator of the Schrödinger
equation. An analytical expression for the time-evolved wave
function is found when the initial state is a Fock-Darwin state.
We also calculate the transition probability to a given FockDarwin state as a consequence of the interaction with the
electromagnetic field. The arrangement of the paper is as
follows: In Sec. 2, the theoretical model is presented. In Sec.
3, the application is performed. Finally, the conclusions are
presented in Sec. 4.
2.
Theoretical Model
In this section we will develop the quantum mechanical formalism to describe the dynamics of electrons in a quantum
dot interacting with a homogeneous time-dependent magnetic field, B (t), and its corresponding induced electric field.
Since the Hamiltonian is time-dependent, energy will not be
conserved and there will not be an energy spectrum. Therefore, the problem must be approached by directly solving the
time-dependent Schrödinger equation:
½
e i2
1 h
P+ A +U
2m
c
¾
Φ (r, t) = i~
∂
Φ (r, t) ,
∂t
(1)
where m is the electron effective mass, q = −e is the electron charge, c is the speed of light and the vector potential
is chosen to be A = −r × B/2. In order to analyze a QD
phenomenon, we will model the dot by a parabolic potential
1/2
whose strength is related to the dot size ld = (~/m$0 ) ,
where $0 is the corresponding oscillator frequency. There-
113
ON THE TRANSITION PROBABILITY FOR A QUANTUM DOT IN A TIME DEPENDENT MAGNETIC FIELD
fore, the potential, U , can be written as:
1
m$02 r2
(2)
2
The temporal evolution of the system from an initial instant t0 to a final instant tf is given by
Z
Φ (rf , tf ) = d2 ri G (rf , tf , ri , t0 ) Φ0 (ri , t0 ) ,
(3)
U (r) =
where G (rf , tf , ri , t0 ) is the propagator and Φ0 (ri , t0 ) is the
initial wave function. We will assume that the magnetic field
has the form B (t) = B0 f (t) ẑ; Hence, the transversal propagator, can be cast in the form:
m
G (rf , tf , ri , t0 ) =
2πi~µ1 (tf )
½
im £
× exp
µ̇1 (tf ) r2f + µ2 (tf ) r2i
2~µ1 (tf )
¾
¤
− 2rf · R (θ (tf )) ri
(4)
where
·
R (θ (t, t0 )) =
cos θ (t, t0 ) − sin θ (t, t0 )
sin θ (t, t0 ) cos θ (t, t0 )
¸
(5)
Rt
with θ (t, t0 ) = t0 ω (t0 ) dt0 and where µ1,2 (t) denote two
linear independent solutions of the equation
¸
· 2
d
+ Ω (t) µ (t) = 0
(6)
dt2
2
with Ω (t) = ω (t) + $02 , such that
ω (t) = (eB0 /(2mc)) f (t) = ωL f (t) .
It is convenient to choose µ1 (t) having dimensions of time
and µ2 (t) being dimensionless, and to choose the initial conditions µ1 (t0 ) = µ̇2 (t0 ) = 0, µ2 (t0 ) = µ̇1 (t0 ) = 1. We
note that, if ωL = 0 or $0= 0 the proper limiting behaviors are
recovered [16–17]. Also, we remark that the properties of the
system generally follow from the behavior of the solutions of
Eq. (6). A particular case of this theoretical model will be
analyzed in the next section.
3.
Application
In this section we will focus in a particular shape of the time
dependence of the magnetic field. We will consider a magnetic field that changes linearly in time, flipping its direction
in a given time interval [−τ, τ ]. The z-component of the magnetic field is−B0 for t ≤ −τ . During the interval under consideration, it linearly increases in time as B (t) = B0 t/τ ,
and fort ≥ τ it becomes +B0 . In order to calculate the corresponding propagator, we must find the general solution of
Eq. (6), that is:
³
´
2
2
µ̈ (t) + $02 + ωL
(t/τ ) µ (t) = 0
(7)
with −τ ≤ t ≤ τ . Let (ζ1 , ζ2 ) be two linearly independent solutions of Eq. (7) satisfying the initial conditions
ζ1 (−τ ) = ζ̇2 (−τ ) = 0 and ζ2 (−τ ) = ζ̇1 (−τ ) = 1. We
can easily express these solutions in terms of the solutions
of Eq. (7) for the initial conditions µ1 (0) = µ̇2 (0) = 0,
µ2 (0) = µ̇1 (0) = 1. The result is:
ζ1 (t) = µ2 (τ ) µ1 (t) + µ1 (τ ) µ2 (t)
(8)
ζ2 (t) = µ̇2 (τ ) µ1 (t) + µ̇1 (τ ) µ2 (t)
(9)
where
¡
¢
µ1 (t) = t exp −iωL t2 /2τ
¡
¢
× 1 F1 i$02 τ /(4ωL )+3/4, 3/2, iωL t2 /τ
¡
¢
µ2 (t) = exp −iωL t2 /2τ
¡
¢
× 1 F1 i$02 τ /(4ωL )+1/4, 1/2, iωL t2 /τ
(10)
(11)
1 F1
(a, b, z)being the usual Hypergeometric function; also,
using the series expansion of this function it can be easily verified that the functions µ1,2 are real, and it is trivially proven
that the functions {µ1 , µ2 }are odd and even functions of t, respectively. Figure 1 shows the time dependence of functions
{ζ1 , ζ2 }in the interval −τ < t < τ for different values of the
dimensionless ratio $0 /ωL , where we have chosen ωL =1 and
τ = 10. Both functions have an oscillating behavior whose
period decreases as the ratio $0 /ωL increases.
Now let us elucidate the structure of the propagator
G (rf , τ, ri , −τ ). Firstly, we choose θ (−τ ) = 0, hence
θ (τ ) = 0; and using the parity of the functions {µ1 , µ2 }
we find that
ζ1 (τ ) = 2µ1 (τ ) µ2 (τ ) ,
ζ2 (τ ) = 2µ̇1 (τ ) µ2 (τ ) − 1,
ζ̇1 (τ ) = ζ2 (τ )
and ζ̇2 (τ ) = 2µ̇1 (τ ) µ̇2 (τ ). Therefore, Eq. (4) becomes
m
4πi~µ1 (τ ) µ2 (τ )
¶¸
µ
·
im ζ2 (τ )
2
(12)
r2f + r2i −
rf · ri
× exp
4~µ1 (τ ) µ2 (τ )
ζ2 (τ )
G (rf , τ, ri , −τ ) =
In what follows we will analyze the temporal evolution of
the QD wave function. The wave function at t = −τ can be
expressed as a superposition of Fock–Darwin states [16–17]:
X
Cl,n exp (iEl,n τ /~) ψl,n (ri )
(13)
Φ (ri , −τ ) =
l,n
where the Cl,n are constants,
µ
¶
q
2
El,n =~$0 (2l+1+ |n|) 1+ (ωL /$0 ) +n (ωL /$0 ) ,
Rev. Mex. Fı́s. S 53 (7) (2007) 112–116
114
D. LAROZE AND R. RIVERA
The integral in Eq. (21) converges if A − Re [Z] > 0, and
in such case its value is:
and
³q
´
ψl,n (ri ) =
pl|n| /πλ2ω
·
¸
q
2
2
2
× exp inφi − mri $0 + ωL /2~
× (ri /λω )
|n|
|n|
³
Ll
Z∞
|n|+1
dri ri
2
´
(ri /λω )
¡
¢
exp −Z (τ ) ri2 Jn (s (τ ) rri )
0
(14)
1
=
2Z (τ )
where pl|n| = l!/(l + |n|)!,
µ
¶1/2
q
2
λω = (~/(m$0 ))/ 1 + (ωL /$0 )
,
µ
¶n
α (τ ) r
ζ2 (τ ) Z (τ )
"
µ
¶2 #
1
α (τ ) r
× exp −
, (22)
Z (τ ) ζ2 (τ )
|n|
Ll (x) are the generalized Laguerre polynomials, and
{ri , φi } are the polar coordinates of ri . For simplicity, we
will analyze the case l = 0. In such scenario the initial wave
function can be cast as
£
¤ |n|
Φ0,n (ri , −τ ) = C̃0,n (τ ) exp inφi − Ari2 ri
(15)
p
2 /2~ and
where A = m $02 + ωL
where, for simplicity, we have assumed that n > 0. Therefore, Eq. (16) can be expressed as:
C̃0,n (τ )
Φ (r, τ ) =
i
× exp
with C0 a normalization constant. Therefore, using Eqs. (3),
(12) and (15), the time-evolved wave function is given by
C̃0,n (τ ) α (τ )
πiζ2 (τ )
£
¤
× exp iα (τ ) r2 ηn (r, ri ; τ )
(16)
where α (τ ) = mζ2 (τ )/(4~µ1 (τ ) µ2 (τ )) and
Z
£
¤ |n|
ηn (r, ri ; τ ) = d2 ri exp inφi − Z (τ ) ri2 ri
× exp [−isr · ri ] (17)
with s (τ ) = 2α (τ )/ζ2 (τ ) and Z (τ ) = A − iα (τ ). Let us
first calculate the angular integral:
dφi exp [inφi − isrri cos (φi − φ)]
Iφ =
1
iα (τ ) −
Z (τ )
¶n+1
µ
r|n|
α (τ )
ζ2 (τ )
#
¶2 )
r
2
(23)
Equation (23) is the exact expression for the time evolved
wave function when the initial wave function is a particular
Fock-Darwin state. We note that it has a similar structure to
the initial wave function, the dissimilarity appearing in the
coefficients and in a numerical phase.
Now let us calculate the probability transition to different
Fock-Darwin states due to the particular time dependence of
the magnetic field. Let us consider the states Φ0,k (−τ ) and
Φ0,n (τ ). The transition amplitude is:
Z
h Φ (−τ )| Φ (τ )i =
Z2π
α (τ )
ζ2 (τ ) Z (τ )
× exp (in (φ − π/2))
"(
C̃0,n (τ ) = C0 exp (iτ E0,n /~)
Φ0,n (r, τ ) =
µ
d2 r Φ∗0,k (r, −τ ) Φ0,n (r, τ )
(24)
(18)
0
Using the integral representation of the Bessel function
[18]:
Jn (ξ) =
1
2π
2π+a
Z
exp [i (nx − ξ sin x)] dx
(19)
Equation (24) can be written as
Z2π
h Φ (−τ )| Φ (τ )i = Xn,k (τ )
dφ exp (iφ (n − k))
0
a
Z∞
Equation (18) can be written as
×
Iφ = 2π exp (in (φ − π/2)) Jn (s (τ ) rri )
(20)
Thus, we finally obtain the following expression for
Eq. (11)
Z∞
|n|+1
ηn (r, ri ; τ ) = 2π exp (in (φ − π/2)) dri ri
¡
× exp −Z
(τ ) ri2
¢
0
Jn (s (τ ) rri ) ,
£
¤
drr|n|+|k|+1 exp −∆ (τ ) r2 (25)
0
with
∗
(τ )
C̃0,n (τ ) C̃0,k
Xn,k (τ ) =
i
(21)
Rev. Mex. Fı́s. S 53 (7) (2007) 112–116
µ
α (τ )
ζ2 (τ ) Z (τ )
¶n+1
× exp (−inπ/2)
(26)
ON THE TRANSITION PROBABILITY FOR A QUANTUM DOT IN A TIME DEPENDENT MAGNETIC FIELD
and
∆ (τ ) = Z (τ ) +
1
Z (τ )
µ
α (τ )
ζ2 (τ )
¶2
(27)
Since
Z2π
dφ exp (iφ (n − k)) = 2πδn,k
(28)
115
the ratio $0 /ωL for the n = 1case. We note that it has a nontrivial structure, with an oscillatory dependence on the length
of the time interval τ , and a peak as a function of $0 /ωL . As
a consequence of these behaviors, the transition probability
presents a line of maximums in certain regions of the parameter space; besides, for a fixed τ , the probability transition
decays asymptotically for large $0 /ωL , away from the line
of maximums.
0
where δn,k is the Kronecker delta, the only possible transition is that which conserves the z-component of the angular
momentum. Hence, Eq. (25) is reduced to
h Φ (−τ )| Φ (τ )i = 2πXn,n (τ )
Z∞
×
¤
£
drr2|n|+1 exp −∆ (τ ) r2
(29)
0
Since
½
α2
Re Z + 2
ζ2 Z
¾
=A+
α2 A
> 0,
ζ22 (α2 + A2 )
the integral converges. Its value is
Z∞
£
¤
drr2|n|+1 exp −∆ (τ ) r2 =
0
|n|!
|n|+1
2 (∆ (τ ))
.
(30)
Therefore,
h Φ (−τ )| Φ (τ )i =
³
¶n+1
¯2 µ
2π ¯¯
m
¯
¯C̃0,n (τ )¯
i
4~µ1 (τ ) µ2 (τ )
|n|! exp (−inπ/2)
2
Z (τ ) + (α (τ )/ζ2 (τ ))
2
Conclusions
The dynamics of electrons in a quantum dot interacting with
a uniform time-dependent magnetic field has been analyzed
when the magnetic field changes linearly in time, flipping its
direction in a given time interval. An exact analytical solution for the time-evolved wave function has been obtained
when the initial state corresponds to a specific Fock-Darwin
level. The classical dynamics of this system has been found;
it presents a non trivial oscillatory behavior through its dependence on the hypergeometric functions in Eq. (12). Both
the amplitude and the period of the oscillation decrease as
the strength of the dot potential, and therefore its confining
properties, increases. The evolved wave function has a structure similar to that of the initial state, but with different timedependent coefficients. In addition, the probability transition
has been calculated for these states; we found that this probability is appreciably different from zero only in some regions
of the parameter space.
Acknowledgments
´|n|+1
(31)
The transition probability is of course proportional to
2
|h Φ (−τ )| Φ (τ )i| . Figure 2 shows the un-normalized transition probability as a function of τ (in arbitrary units) and
∗
4.
. Corresponding Author: Tel. +56-32-273136; Fax +56-32273529; e-mail [email protected].
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