A new approach to Ermakov systems and applications in quantum physics
Jos´ e F. Cari˜ nena University of Zaragoza
[email protected] CEWQO 2007, June 5, 2007
Abstract Milne-Pinney equation x ¨ = −ω 2 (t)x + k/x3 is usually studied together with the time-dependent harmonic oscillator y¨ + ω 2 (t)y = 0 and the system is called Ermakov system, and actually Pinney showed in a short paper that the general solution of the first equation can be written as a superposition of two solutions of the associated harmonic oscillator. A recent generalization of the concept of Lie systems for second order differential equations and the usual techniques of Lie systems will be used to study the Ermakov system. Several applications of Ermakov systems in Quantum Mechanics as the relation between Schroedinger and Milne equations or the use of Lewis–Riesenfeld invariant will be analysed from this geometric viewpoint.
Keywords: Ermakov systems, Milne–Pinney equation, superposition rules, Lie systems.
1
Outline
1. Non-autonomous systems of differential equations 2. Reduction process for a 2-dim isotropic harmonic oscillator 3. Time-independent Schrodinger equation 4. Lie systems: A geometric approach 5. SODE Lie systems 6. Application in Bose–Einstein condensate 7. Propagation of Gaussian beams in nonlinear media
2
Non-autonomous systems The interest of the h.o. with a t-dependent angular frequency ω(t) is beyond doubt: x˙ = v 2 x ¨ = −ω (t) x ⇐⇒ v˙ = −ω 2 (t) x whose solutions are the integral curves of the t-dependent vector field ∂ ∂ − ω 2 (t)x . ∂x ∂v This is a simple example of a nonautonomus (system of) differential equation(s). X=v
There is no general method for finding the general solution (or even a particular one). For the particular case of the h.o. (the linear case): If we know a particular solution the general solution can be found by a quadrature, If we know two particular solutions, x1 and x2 , the general solution is a linear combination (no quadrature is needed) x(t) = k1 x1 (t) + k2 x2 (t) 3
There are other systems for which the general solution can be written as a (maybe nonlinear) function of some particular solutions. For instance, for Riccati equation: If a particular solution is known, the general solution is obtained by two quadratures, If two particular solutions are known the problem reduces to one quadrature and When three particular solutions are known, x1 , x2 and x3 , the general solution can be found from the cross ratio relation x − x1 x3 − x1 : =k , x − x2 x3 − x2 which provides us a nonlinear superposition rule. We aim to study another nonlinear differential equation ( x˙ = v k 2 x ¨ = −ω (t)x + 3 , k x v˙ = −ω 2 (t)x + 3 x which determines the integral curves of the t-dependent vector field ∂ k ∂ 2 X=v + −ω (t)x + 3 . ∂x x ∂v 4
Reduction process for a 2-dim isotropic harmonic oscillator Starting with an isotropic two-dimensional harmonic oscillator with a time dependent frequency ω(t), described by x ¨1 = −ω 2 (t)x1 x ¨2 = −ω 2 (t)x2 and using polar coordinates, for instance with z = x1 + i x2 = ρ eiθ , we see that ˙ , z¨ = eiθ [¨ ρ − ρ θ˙2 + i(ρ θ¨ + 2 ρ˙ θ)] and therefore the equations of motion become ρ¨ − ρ θ˙2 + ω 2 (t) ρ = 0 1 d 2 ˙ ρ θ =0 ρ dt The second equation shows that ` = ρ2 θ˙ is a constant of the motion and using such constant the first equation is transformed into `2 ρ¨ − ρ θ˙2 + ω 2 (t) ρ = 3 , ρ 5
usually called Milne equation. In other words, such a differential equation appears as a reduction from the isotropic two-dimensional harmonic oscillator when taking into account the constancy of the angular momentum. Note also that z ∗ z˙ = ρe−iθ ρ˙ eiθ + i θ˙ ρ eiθ and therefore Im (z ∗ z) ˙ = ρ2 θ˙ . Moreover, since z ∗ z˙ = (x1 − i x2 )(x˙ 1 + i x2 ) = x1 x˙ 1 + x2 x˙ 2 + (x1 x˙ 2 − x˙ 1 x2 ) we see that the constant of motion Im (z ∗ z) ˙ is nothing but the Wronskian of the two solutions of the harmonic oscillator. Moreover, if instead of using two solutions (x1 , x2 ) we use another solution obtained by a rotation x01 = cos α x1 − sin α x2 ,
x02 = sin α x1 + cos α x2 6
then z becomes z 0 = z eiα , and therefore, as Im (z 0∗ z˙ 0 ) = Im (z ∗ z) ˙ ,
|z 0 | = |z| ,
we see that the solutions (x01 , x02 ) of the harmonic oscillator produce the same solution of the corresponding Milne equation with the same parameter `. Morever, if we want to determine the solutions obtained by means of this procedure, we shuld consider two particular solutions with a Wronskian W (x1 , x2 ) and two copies of the general solution A1 x1 + A2 x2 , B1 x1 + B2 x2 . Then the Wronskian is given by W (A1 x1 + A2 x2 , B1 x1 + B2 x2 ) = (A1 B2 − A2 B1 )W (x1 , x2 ) . The two general solutions we have considered give rise to p ρ = (A1 x1 + A2 x2 )2 + (B1 x1 + B2 x2 )2 i.e. ρ=
q
(A21 + B12 )x21 + (A22 + B22 )x22 + 2(A1 A2 + B1 B2 )x1 x2
as a solution of of the Milne–Pinney equation with k = (A1 B2 − A2 B1 )W (x1 , x2 ). 7
Remark that such a solution depends only on the invariants under rotations of the vectors M1 = (A1 , B1 ) and M2 = (A2 , B2 ), with the mentioned additional restriction, also invariant under rotations. Conversely, if ρ is a particular solution and define the function θ(t) by the quadraturae Z θ(t) =
t
1 dt0 , ρ2 (t0 )
then x1 +ix2 = ρ eiθ determines a solution for the bi-dimensional harmonic oscillator: Z t 1 0 x = ρ cos dt + α . ρ2 (t0 )
8
Time-independent Schrodinger equation The time-independent Schroedinger equation for a one-dimensional quantum system is given by d2 ψ 2m + 2 (E − V (x))ψ(x) = 0 , dx2 ~ which can be written as d2 ψ + k 2 (x)ψ(x) = 0 , dx2 to be compared with the equation in complex form for the classical motion of the two-dimensional harmonic oscillator with a time-dependent angular frequency ω(t). x ψ k(x) Hilbert space
←→ ←→ ←→ ←→
t x1 + i x2 ω(t) C
We can therefore proceed in the same way as in the finite-dimensional case.
9
Assume the factorization ψ(x) = A(x) eiS(x) (amplitud-phase approach), where A and S are real functions such that A(x) ≥ 0. The Schroedinger equation is equivalent to A00 (x) + [k 2 (x) − S 0 (x)]A = 0 , 1 d A2 (x)S 0 (x) = 0 A(x)S 00 (x) + 2S 0 (x)A0 (x) = A(x) dx The second equation implies the existence of a constant κ such that A2 (x) S 0 (x) = κ and using this value for S 0 (x) in terms of A(x) in the first equation we find that A satisfies κ A00 (x) + k 2 (x) A(x) − 3 = 0. A (x) If a particular solution of such equation is known, the general solution of the Schroedinger equation can be written: Z t 1 0 ψ(x)) = A(x) cos dx + α . A2 (x0 ) 10
Lie systems: A geometric approach Consider a nonautonomous system time-dependent first order differential equations dxi (t) = X i (x, t) , dt
i = 1, . . . , n.
From the geometrical viewpoint, this system determines the integral curves of X(x, t) =
n X i=1
X i (x, t)
∂ , ∂xi
i = 1, . . . , n.
and defines at each point of M a linear subspace (the rank may be non constant), i.e. it determines a ‘generalized’ distribution. The simplest example would be the homogeneous linear system n
X dxi = Ai j (t) xj , dt j=1
i = 1, . . . , n,
for which the general solution can be written as a linear combination x(t) = c1 x(1) + · · · + cn x(n) 11
of n independent particular solutions. For the inhomogeneous linear system n
X dxi = Ai j (t) xj + ai (t) , dt j=1
i = 1, . . . , n ,
the general solution is a linear combination of (n + 1) independent particular solutions:: x(t) = x(1) + c2 (x(2) − x(1) ) + · · · + cn+1 (x(n+1) − x(n) ) . If we perform a (non-linear) change of variable, y = φ(x), then the given linear system will become nonlinear but the property of admitting a superposition function should remain. Let us look for systems admitting a (may be nonlinear) superposition rule. What about the superposition function and the number m of solutions?
12
Lie theorem Theorem Given a non-autonomous system of n first order differential equations dxi = X i (x1 , . . . , xn , t), dt
i = 1 . . . , n,
a necessary and sufficient condition for the existence of a function Φ : Rn(m+1) → Rn such that the general solution is x = Φ(x(1) , . . . , x(m) ; k1 , . . . , kn ) , with {x(a) | a = 1, . . . , m} being a set of particular solutions of the system and k1 , . . . , kn , are n arbitrary constants, is that the system can be written as dxi = Z 1 (t)ξ1i (x) + · · · + Z r (t)ξri (x), dt where Z 1 , . . . , Z r , are r functions depending only on t and ξαi , α = 1, . . . , r, are functions of x = (x1 , . . . , xn ), such that the r vector fields in Rn given by Xα ≡
n X i=1
ξαi (x1 , . . . , xn )
∂ , ∂xi
α = 1, . . . , r, 13
close on a real finite-dimensional Lie algebra, i.e. the Xα are l.i. and there are r3 real numbers, cαβ γ , such that [Xα , Xβ ] =
r X
cαβ γ Xγ .
γ=1
The number r satisfies r ≤ mn.
The condition in the Theorem is that X(x, t) can be written as X(x, t) =
r X
Z α (t)Xα (x) ,
α=1
with Xα as mentioned, i.e. such that [Xα , Xβ ] =
r X
cαβ γ Xγ .
γ=1
Non-autonomous systems corresponding to such vector fields along π will be called Lie systems 14
Examples A) Linear systems n
X dxi = Ai j (t) xj . dt j=1 In this case m = n and the linear superposition principle is given by Φ(x(1) , . . . , x(n) ; k1 , . . . , kn ) = k1 x(1) + · · · + kn x(n) . There are n2 vector fields Xij = xj
∂ , ∂xi
which close on the gl(n, R) algebra. ∂ ∂ l ∂ j ∂ ,x = δ il xj k − δ kj xl i [Xij , Xkl ] = x ∂xi ∂xk ∂x ∂x i.e. [Xij , Xkl ] = δ il Xkj − δ kj Xil , to be compared with the commutation relations of the gl(n, R) algebra [Eij , Ekl ] = δjk Eil − δil Ekj . with (Eij )kl = δik δjl . 15
B) For inhomogeneous systems, the time-dependent vector field is n n X X ∂ X= Ai j (t) xj + B i (t) i , ∂x i=1 j=1 which is a linear combination with t-dependent coefficients, X=
n X
Ai j (t) Xij +
i,j=1
n X
B i (t) Xi ,
i=1
of the n2 vector fields Xij and the n vector fields Xi =
∂ , ∂xi
i = 1, . . . , n .
Now, the commutation relations of these vector fields are [Xi , Xk ] = 0 ,
[Xij , Xk ] = −δkj Xi ,
∀ i, j, k = 1, . . . , n .
The Lie algebra generated by the vector fields {Xij , Xk | i, j, k = 1, . . . , n} is isomorphic to the (n2 + n)-dimensional Lie algebra of the affine group. In this case r = n2 + n and m = n + 1 and the equality r = m n also follows. 16
C) The Riccati equation dx(t) = a2 (t) x2 (t) + a1 (t) x(t) + a0 (t) . dt Now m = 3 and the superposition principle comes from the relation x − x1 x3 − x1 : =k , x − x2 x3 − x2 or in other words, x=
k x1 (x3 − x2 ) + x2 (x1 − x3 ) . k (x3 − x2 ) + (x1 − x3 )
The vector fields X0 , X1 and X2 are given by X0 =
∂ , ∂x
X1 = x
∂ , ∂x
X2 = x2
∂ , ∂x
that close on a three-dimensional real Lie algebra, with defining relations [X0 , X1 ] = X0 ,
[X0 , X2 ] = 2X1 ,
[X1 , X2 ] = X2 ,
i.e. the sl(2, R) algebra. 17
SODE Lie systems A system of second order differential equations x ¨i = f i (t, x, x) ˙ ,
i = 1, . . . , n,
can be studied through the corresponding system of first order differential equations i dx = v i dti dv = f i (t, x, v) dt with associated t-dependent vector field X = vi
∂ ∂ + f i (t, x, v) i i ∂x ∂v
We call SODE Lie systems those for which X is a Lie system, i.e. it can be written as a linear combination with t-dependent coefficients of vector fields closing a finitedimensional real Lie algebra. 18
Examples A) The 1-dim harmonic oscillator with time-dependent frequency The equation of motion is x ¨ = −ω 2 (t)x with associated system
x˙ = v v˙ = −ω 2 (t)x
and vector field
∂ ∂ − ω 2 (t)x , ∂x ∂v which is a linear combination X = X2 − ω 2 (t)X1 with X=v
X1 = x such that if X3 =
∂ , ∂v 1 2
x
X2 = v ∂ ∂ −v ∂x ∂v
∂ , ∂x
.
then [X1 , X2 ] = 2 X3 ,
[X1 , X3 ] = −X1 ,
[X2 , X3 ] = X2 ,
a Lie algebra isomorphic to sl(2, R). This system has no first integrals. 19
B) The 2-dim isotropic harmonic oscillator with time-dependent frequency The equation of motion is
x ¨1 x ¨2
= −ω 2 (t)x1 = −ω 2 (t)x2
with associated system x˙ 1 v˙ 1 x ˙2 v˙ 2
= = = =
v1 −ω 2 (t)x1 v2 −ω 2 (t)x2
and the vector field X = v1
∂ ∂ ∂ ∂ − ω 2 (t)x1 + v2 − ω 2 (t)x2 , ∂x1 ∂v1 ∂x2 ∂v2
is a linear combination X = X2 − ω 2 (t)X1 with X1 = x1 such that if
∂ ∂ + x2 , ∂v1 ∂v2
1 X3 = 2
X2 = v1
∂ ∂ + v2 , ∂x1 ∂x2
∂ ∂ ∂ ∂ x1 − v1 + x2 − v2 ∂x1 ∂v1 ∂x2 ∂v2
. 20
then [X1 , X2 ] = 2 X3 ,
[X1 , X3 ] = −X1 ,
[X2 , X3 ] = X2 ,
once again a Lie algebra isomorphic to sl(2, R). The system admits an invariant because, if F is given by F (x1 , x2 , v1 , v2 ), then X2 F = 0 shows that there exists a function F¯ (ξ, v1 , v2 ) with ξ = x1 v2 − x2 v1 , such that F (x1 , x2 , v1 , v2 ) = F¯ (ξ, v1 , v2 ) while the condition X1 F = 0 ∂ F¯ ∂ F¯ + x2 =0 x1 ∂v1 ∂v2 i.e. we obtain the first integral F = x1 v2 − x2 v1 which can be seen as a partial superposition rule. Actually, if x1 (t) is a solution of the first equation, then we obtain for each real number k the first-order differential equation for the variable x2 dx2 x1 (t) = k + x˙ 1 (t)x2 , dt from where x2 can be found to be given by Z t dζ 0 x2 (t) = k x1 (t) + k x1 (t) . x21 (ζ) 21
With three copies of the same harmonic oscillator, the vector fields X1 and X2 are X1 = v1
∂ ∂ ∂ + v2 +v , ∂x1 ∂x2 ∂x
X2 = x1
∂ ∂ ∂ + x2 +x , ∂v1 ∂v2 ∂v
which determine the first integrals F as solutions of X1 F = X2 F = 0. The condition X1 F = 0 says that there exists a function F¯ : R5 → R2 such that F (x1 , x2 , x, v1 , v2 , v) = F¯ (ξ1 , ξ2 , v1 , v2 , v) with ψ1 (x1 , x2 , x, v1 , v2 , v) = xv1 − x1 v and ψ2 (x1 , x2 , x, v1 , v2 , v) = xv2 − x2 v, and the condition X2 F = 0 transforms into x1
∂ F¯ ∂ F¯ ∂ F¯ + x2 + x1 ∂v1 ∂v2 ∂v
i.e. ξ1 and ξ2 are first integrals. They produce a superposition rule, because xv2 − x2 v = k1 x1 v − v1 x = k2 from where we obtain the expected superposition rule: x = k1 x1 + k2 x2 ,
v = k1 v1 + k2 v2 .
22
C) Pinney equation: The Pinney equation is the following second order non-linear differential equation: k , x3 where k is a constant. The corresponding system of first order differential eqs is x˙ = v v˙ = −ω 2 (t)x + xk3 x ¨ = −ω 2 (t)x +
and the associated vector field ∂ k ∂ 2 X=v + −ω (t)x + 3 . ∂x x ∂v This is a Lie system because it can be written as X = L2 − ω 2 (t)L1 , where :
∂ k ∂ ∂ , L2 = 3 +v . ∂v x ∂v ∂x The vector fields L1 and L2 span a three-dimensional real Lie algebra g with nonzero defining relations: L1 := x
[L1 , L2 ] = 2L3 ,
[L3 , L2 ] = −L2 ,
[L3 , L1 ] = L1 23
where
1 L3 = 2
∂ ∂ −v x ∂x ∂v
,
which is isomorphic to sl(2, R). The fact that they have the same associated Lie algebra means that they can be solved simultaneously in the group SL(2, R) by the equation g˙ g −1 = ω 2 (t) a1 − a2 Note that this isotonic oscillator shares with the harmonic one the property of having a period independent of the energy, i.e. they are isochronous, and in the quantum case they have a equispaced spectrum.
24
D) Ermakov system Consider the system x˙ = v˙ x = y ˙ = v˙ = y
vx −ω 2 (t)x vy −ω 2 (t)y +
1 y3
with associated vector field ∂ ∂ 1 ∂ ∂ 2 2 + vy − ω (t)x + −ω (t)y + 3 , X = vx ∂x ∂y ∂vx y ∂vy which is a linear combination with time-dependent coefficients of the vector fields, X = −ω 2 (t)X1 + X2 , of the vector fields X1 = x
∂ ∂ +y , ∂vx ∂vy
X2 = vx
∂ ∂ 1 ∂ + vy + 3 . ∂x ∂y y ∂vy
This system is made up by two Lie systems closing on a sl(2, R) algebra. The first equation is a h.o. and the second first order differential equation subsystem corresponds to the Pinney equation. The generators of the Lie system with algebra sl(2, R) span a distribution of dimension two and ther is no first integral of the motion 25
for such subsystem. By adding the other sl(2, R) linear Lie system, the harmonic oscillator with time dependent angular frequency, as the distribution in the fourdimensional space is of rank three there is an integral of motion. The first integral can be obtained from X1 F = X2 F = 0. But X1 F means that F (x, y, vx , vy ) = F¯ (x, y, ξ) with ξ = xvy − yvx , and then X2 F = 0 is written vx
x ∂ F¯ ∂ F¯ ∂ F¯ + vx + 3 ∂x ∂x y ∂ξ
and rmthe associated charactristics system we obtain x dy − y dx y 3 dξ d(x/y) y dξ = =⇒ + =0 ξ x ξ x from where and the fiollowing first integral is found: 2 2 x x 2 +ξ = + (xvy − yvx )2 ψ(x, y, vx , vy ) = y y which is the well-known Ermakov invariant.
26
E) Generalized Ermakov system It is the system given by: x ¨ = y¨ =
1 f (y/x) − ω 2 (t)x x3 1 g(y/x) − ω 2 (t)y y3
In the particular case f (u) = 0 and g(u) = 1 reduces to the Ermakov system. This system can be written as a first order one by doubling the number of degrees of freedom by introducing the new variables vx and vy : x˙ = vx v˙ = −ω 2 (t)x + 1 f (y/x) x
y˙ v˙ y
x3
= vy = − ω 2 (t)y +
1 y 3 g(y/x)
which determines the integral curves of the vector field ∂ 1 ∂ 1 ∂ ∂ X = vx +vy + −ω 2 (t)x + 3 f (y/x) + −ω 2 (t)y + 3 g(y/x) . ∂x ∂vy x ∂vx y ∂vy Such vector field can be written as a linear combination X = N2 − ω 2 (t) N1 27
where N1 and N2 are the vector fields N1 = x
∂ ∂ +y , ∂vx ∂vy
N2 = vx
1 ∂ 1 ∂ ∂ ∂ + f (y/x) + vy + g(y/x) , ∂x x3 ∂vx ∂y y 3 ∂vy
Note that these vector fields generate a three-dimensional real Lie algebra with a third generator 1 ∂ ∂ ∂ ∂ N3 = x − vx +y − vy . 2 ∂x ∂vx ∂y ∂vy In fact, as [N1 , N2 ] = 2N3 ,
[N3 , N1 ] = N1 ,
[N3 , N2 ] = −N2
they generate a Lie algebra isomorphic to sl(2, R). Therefore the system is a Lie system.
28
There exists a first integral for the motion, F : R4 → R, for any ω 2 (t), because this Lie system has an associated integrable distribution of rank three and the manifold is four-dimensional. This first integral F satifies Ni F = 0 for i = 1, . . . , 3, but as [N1 , N2 ] = 2N3 it is enough to impose N1 F = N2 F = 0. Then, if N1 F = 0, x
∂F ∂F +y = 0, ∂vx ∂vy
and according to the method of characteristics we obtain: dy dvx dvy dx = = = 0 0 x y and therefore there exists a function F¯ : R3 → R such that F (x, y, vx , vy ) = F¯ (x, y, ξ = xvy − yvx ). The condition N2 F = 0 reads now ¯ ∂ F¯ ∂ F¯ y x ∂F vx + vy + − 3 f (y/x) + 3 g(y/x) . ∂x ∂y x y ∂ξ We can therefore consider the associated system the characteristics are given by: dx dy dξ = = y vx vy − x3 f (y/x) +
x y 3 g(y/x) 29
But using that −y dx + x dy dx dy = = ξ vx vy we arrive to
dξ −y dx + x dy = ξ − xy3 f ( xy ) +
i.e. y2 d −
y x y 3 g( x )
x y
ξ
=
dξ − xy3 f ( xy ) +
y x y 3 g( x )
and integrating we obtain 1 2 ξ + 2
Z
x/y
1 f u3
1 1 + ug du . u u
This first integral allows us to determine a solution of one subsystem in terms of a solution of the other equation.
30
F) The Pinney equation revisited Consider the system of first order differential equations: x˙ = vx y˙ = vy z˙ = v z
v˙ x v˙ y v˙ z
= −ω 2 (t)x = −ω 2 (t)y + = −ω 2 (t)z
k y3
which corresponds to the vector field ∂ ∂ k ∂ ∂ ∂ ∂ ∂ 2 + vy + vz + 3 − ω (t) x +y +z X = vx ∂x ∂y ∂z y ∂vy ∂vx ∂vy ∂vz The vector field X can be expressed as X = N2 − ω 2 (t)N1 where the vector fields N1 and N2 are: N1 = y
∂ ∂ ∂ +x +z , ∂vy ∂vx ∂vz
N2 = vy
1 ∂ ∂ ∂ ∂ + 3 + vx + vz , ∂y y ∂vy ∂x ∂z
These vector fields generate a three-dimensinal real Lie algebra with the vector field N3 given by 1 ∂ ∂ ∂ ∂ ∂ ∂ N3 = x − vx +y − vy +z − vz . 2 ∂x ∂vx ∂y ∂vy ∂z ∂vz 31
In fact, as [N1 , N2 ] = 2N3 ,
[N3 , N1 ] = N1 ,
[N3 , N2 ] = −N2
they generate a Lie algebra isomorphic to sl(2, R). The system is a Lie system. The distribution generated by these fundamental vector fields has rank three. Thus, as the manifold of the Lie system is of dimension six we obtain three time-independent integrals of motion. The Ermakov invariant I1 of the subsystem involving variables x and y. The Ermakov invariant I2 of the subsystem involving variables y and z The Wronskian W of the subsystem involving variables x and z has They define a foliation with three-dimensional leaves. We can use this foliation for obtaining a superposition rule in terms of these three first integrals.
32
The Ermakov invariants read as: I1
=
1 2
I2
=
1 2
(yvx − xvy )2 + c (yvz − zvy )2 + c
2 x y
2 z y
and W is: W = x1 vvz − zvx In terms of these three integrals we can obtain an explicit expression of y in terms of x, z and the integrals I1 , I2 , W : y=
1/2 p 2 2 I2 x + I1 z 2 ± 4I1 I2 − cw2 xz W
This can be interpreted as saying that there is a superposition rule allowing us to express the general solution of the Pinney equation in terms of two independent solutions of the corresponding harmonic oscillator with time-dependent frequency
33
Application in Bose–Einstein condensate The macroscopic wave function of the Bose–Einstein condensate is determined by the Gross–Pitaevskii equation i~
∂Ψ ~2 2 =− ∇ Ψ + V (x, t)Ψ + g|Ψ|2 Ψ . ∂t 2m
In the two-dimensional case it is possible to determine the dynamical evolution by using the moment method. Such momenta are defined by Z I1 (t) = d2 x|Ψ|2 Z I2 (t) = d2 x r2 |Ψ|2 Z ∂Ψ∗ ∂Ψ∗ I3 (t) = d2 x r Ψ − Ψ∗ ∂r ∂r Z 1 2 2 4 I4 (t) = d x |∇Ψ| + g|Ψ| 2 I1 is constant and the other integral parameters satisfy the system of coupled first 34
order differential equations: dI2 = I3 dt dI3 = −2 ω 2 (t)I2 + 4I4 dt dI4 = − 1 ω 2 (t)I3 dt 2
.
For simplicity we denote yk instead of Ik . The system gives the integral curves of the t-dependent vector field X = X1 − ω 2 (t)X2 ,
X 1 = y3
∂ ∂ + 4 y4 , ∂y2 ∂y3
X2 = −2 y2
1 ∂ ∂ − y3 . ∂y3 2 ∂y4
The vector fields X1 and X2 are such that ∂ ∂ [X1 , X2 ] = 2 X3 = 2 −y3 + y4 . ∂y3 ∂y4 It is easy to check that [X1 , X3 ] = −X1 ,
[X2 , X3 ] = X2 . 35
Therefore the given system is a Lie system with associated Lie algebra sl(2, R). For instance, a Lie isomorphism is given by ϕ(X1 ) = Y0 ,
ϕ(X2 ) = Y2 ,
ϕ(X3 ) = Y1 ,
and therefore the given system corresponds to the Riccati equation y˙ = 1 − ω 2 (t) y 2 .
On the other hand, one can find a first integral by solving the system of differential equations X 1 F = y3
∂F ∂F + 4 y4 = 0, ∂y2 ∂y3
X2 F = −2 y2
∂F 1 ∂F − y3 . ∂y3 2 ∂y4
A solution of the first equation is obtained from the associated system dy2 dy3 dy4 = = y3 4y4 0 i.e. F is a function of F (y2 , y3 , y4 ) = Φ(u, v), where u(y2 , y3 , y4 ) = 4y4 y2 −(1/2)y32 and v = y4 , and in order to be a solution of the second equation, F can only be 36
any function of u. In other words any first integral of the system is a function of u(y2 , y3 , y4 ). If we fix the value of the first integral as 2y4 y2 − y32 /4 = k =⇒ y4 = (1/2y22 )(k + y32 /4) , then when we put y2 = ρ2 , the first equation says that y3 = 2ρρ, ˙ and using this expresion in the second equation, it becomes 1 ρ¨ ρ + ρ˙ 2 = −ω 2 ρ2 + 2 k + ρ˙ 2 ρ and therefore ρ is a solution of the equation: ρ¨ + ω 2 ρ =
k . ρ3
In another recent paper (Herring et al, cond-mat/0701756) the same problem was analysed and the following equation for the momentum y = I2 was found 1 µ y˙ + 4λ y = , y¨ − 2y y which transforms under the change of variables y = Y 2 into µ Y¨ + 2 λY = 2Y 3 37
Propagation of Gaussian beams in nonlinear media Consider a linear medium where the permitivity is given by = 0 (1 − γ(x2 + y 2 ) + β|E|2 ) . The light propagation is described by the nonlinear Schroedinger-like equation −2ik
∂U ∂2U ∂2U + + − k 2 γ(x2 + y 2 )U + k‘2β|U |2 U = 0 . ∂z ∂x2 ∂y 2
where E = U eikz . If the field is given in the boundary of the nonliear medium (z = 0) by 2 x y2 kxy , U (x, y, z) = U0 exp − 2 − 2 − i a0 b0 2R0 we can look for a solution of the form: 2 x y2 xy kxy kx2 ky 2 E = Em exp − 2 − 2 − −i −i −i − iP . a b d 2Rxy 2Rx 2Ry where the coefficients are functions of z. 38
After some computations one find the following subset of equations (Goncharenko et al, 1991) f (b/a) a ¨ + ω2 a = a2 b f (a/b) ¨b + ω 2 b = ab2 which is a particular case of the generalized Ermakov systems.
39
Bibliography [1] Cari˜ nena J.F., Grabowski J. and Marmo G., Lie–Scheffers systems: a geometric approach, Bibliopolis, Napoli, 2000. [2] Cari˜ nena J.F., Grabowski J. and Marmo G., Some applications in physics of differential equation systems admitting a superposition rule, Rep. Math. Phys. 48 (2001) 47–58. [3] Cari˜ nena J.F., Grabowski J. and Marmo G., Superposition rules, Lie theorem and partial differential equations, Preprint 2006, math-ph/0610013 [4] Cari˜ nena J.F., Grabowski J. and Ramos A., Reduction of time–dependent systems admitting a superposition principle, Acta Appl. Math. 66 (2001) 67–87. [5] Cari˜ nena J.F., de Lucas J. and Ra˜ nada M.F., Nonlinear superpositions and Ermakov systems. To appear in Differential Geometric Methods in Mechanics and Field Theory, eds F. Cantrijn, M. Crampin and B. Langerock, Academia Press (2007) 40
[6] Cari˜ nena J.F., Marmo G. and Nasarre J., The nonlinear superposition principle and the Wei–Norman method, Int. J. Mod. Phys. A 13 (1998) 3601–27. [7] Cari˜ nena J.F. and Ramos A., Integrability of the Riccati equation from a group theoretical viewpoint, Int. J. Mod. Phys. A 14 (1999) 1935–51. [8] Cari˜ nena J.F. and Ramos A., Riccati equation, the factorization method and shape invariance, Rev. Math. Phys. 12 (1999) 1279–304. [9] Cari˜ nena J.F. and Ramos A., A new geometric approach to Lie systems and physical applications, Acta Appl. Math. 70 (2002) 43–69. [10] Cari˜ nena J.F. and Ramos A., Lie systems and Connections in fibre bundles: Applications in Quantum Mechanics, 9th Int. Conf. Diff.Geom and Appl., p. 437– 52 (2004), J. Bures et al eds, Matfyzpress, Praga 2005 [11] Herring G., Kevrekidis P.G., Williams F., Christodoulakis T. and Frantzeskakis D.J., From Fesbach-Resonance managed Bose–Einstein Condensates of asisotropic universes: applications of the Ermakov–Pinney equations with time–dependent nonlinearity, cond-mat/0701756
41
[12] Lidsey J.E., Cosmic dynamics of Bose–Einstein condensates, Class. Quantum Grav. 21, 777–85 (2004). [13] Lie S., Vorlesungen u ¨ber continuierliche Gruppen mit Geometrischen und anderen Anwendungen, Edited and revised by G. Scheffers, Teubner, Leipzig, 1893. [14] Pinney E., The nonlinear differential equation y 00 + p(x)y 0 + cy −3 = 0, Proc. A.M.S. 1, 681 (1950)
42
