SEMICLASSICAL TIME-INDEPENDENT...
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SEMICLASSICAL TIME-INDEPENDENT DESCRIPTION OF RAPIDLY OSCILLATING FIELDS ON A LATTICE IN THE KAPITZA APROXIMATION Luis Alfredo Martínez 1 y Jean Pierre Gallinar 1 1 Universidad Simón Bolívar (Venezuela) [email protected] We investigate a semiclassical dynamics driven by a high-frequency (ω) field, plus a static arbitrary potential on a one-dimensional tight-binding lattice. We find –in the spirit of the Kapitza pendulum- an effective, time-independent potential U eff (x) that describes the average motion to order ω -2 . This effective potential depends on the static external potential U(x), on the lattice constant “a” and on the applied high-frequency field f(x, t). One obtains that 1 2 2 2 2 4 * 2 2 2 ) ( ) ( ) ( ) ( ) ( ) ( 2 ) ( n n eff n x f x t E x U dx a t E x U a x U a m x U With ) ( cos 2 ) ( 0 0 x U a k t t E a t t 0 * cos and 1 2 2 0 2 2 0 ) ( 1 cos n n n x f a a Where “m” and “E” are, respectively, the effective mass and unperturbed energy of the particle´s average motion, given by initial conditions (x 0 , k 0 ), f n (x) is the n-th Fourier component of the driving field, “t” is proportional to gap band and finally, a cos is the reduction factor of effective gap band “t * ” imposed by external driven field. As many semiclassical electronic motion on the lattice as well atomic motion in an optical one, are possible applications of our formula for effective potential U eff (x). Keywords: rapid forcing, Kapitza pendulum, effective potential, semiclassical approach. RIAO OPTILAS Code Topics: 10
Transcript of SEMICLASSICAL TIME-INDEPENDENT...
SEMICLASSICAL TIME-INDEPENDENT DESCRIPTION OF RAPIDLY
OSCILLATING FIELDS ON A LATTICE IN THE KAPITZA APROXIMATION
Luis Alfredo Martínez1 y Jean Pierre Gallinar
1
1Universidad Simón Bolívar (Venezuela)
We investigate a semiclassical dynamics driven by a high-frequency (ω) field, plus a
static arbitrary potential on a one-dimensional tight-binding lattice. We find –in the
spirit of the Kapitza pendulum- an effective, time-independent potential Ueff(x) that
describes the average motion to order ω-2
. This effective potential depends on the static
external potential U(x), on the lattice constant “a” and on the applied high-frequency
field f(x, t). One obtains that
1
22
224
*22
2 )()()()()()(
2
)(
n
neff
n
xf
xtExUdxatExUaxU
a
m
xU
With
)(cos2)( 00 xUakttE att 0* cos and
1
22
02
20
)(1cos
n
n
n
xfaa
Where “m” and “E” are, respectively, the effective mass and unperturbed energy of the
particle´s average motion, given by initial conditions (x0, k0), fn(x) is the n-th Fourier
component of the driving field, “t” is proportional to gap band and finally, acos is the
reduction factor of effective gap band “t*” imposed by external driven field. As many
semiclassical electronic motion on the lattice as well atomic motion in an optical one,
are possible applications of our formula for effective potential Ueff(x).
Keywords: rapid forcing, Kapitza pendulum, effective potential, semiclassical
approach.
RIAO OPTILAS Code Topics: 10
DESCRIPCIÓN SEMICLÁSICA INDEPENDIENTE DEL TIEMPO DE
CAMPOS RÁPIDAMENTE OSCILANTES EN LA RED, BAJO LA
APROXIMACIÓN DE KAPITZA
Luis Alfredo Martínez1 y Jean Pierre Gallinar
1
1Universidad Simón Bolívar (Venezuela)
Se investigó la dinámica semiclásica forzada por un campo de alta frecuencia (ω), más
un potencial estático arbitrario sobre un retículo unidimensional. Se encontró -en el
espíritu del péndulo de Kapitza- un potencial efectivo Ueff(x) independiente del tiempo
que describe el movimiento hasta el orden de ω-2
. Este potencial efectivo depende del
potencial estático externo U(x), de la constante de la red “a” del retículo y del campo
oscilante de alta frecuencia f(x, t). Se obtiene que:
1
22
224
*22
2 )()()()()()(
2
)(
n
neff
n
xf
xtExUdxatExUaxU
a
m
xU
con
)(cos2)( 00 xUakttE att 0* cos and
1
22
02
20
)(1cos
n
n
n
xfaa
Donde “m” y “E” son, respectivamente, la masa efectiva y la energía no perturbada del
movimiento promedio de la partícula, dadas por las condiciones iniciales (x0, k0), fn(x) es
la n-ésima componente de Fourier del campo oscilante, “t” es proporcional al ancho de
banda y finalmente, acos es el factor de reducción del ancho de banda efectivo “t*”
impuesto por el campo oscilante externo. Tanto el movimiento electrónico semiclásico
en un retículo cristalino como también el atómico en un retículo óptico, constituyen
posibles aplicaciones de la fórmula obtenida para el potencial efectivo Ueff(x).
Palabras claves: fuerza rápidamente oscilante, péndulo de Kapitza, potencial efectivo,
aproximación semiclásica.
RIAO OPTILAS Código de tópico: 10
