Download Appendix D Wigner Function Formulation of Gross

Appendix D
Wigner Function Formulation of
Gross-Pitaevskii Dynamics
D.1
Overview
The goal here is to derive a closed equation of motion for the Wigner function, when the
corresponding wavefunction dynamics are governed by the Gross-Pitaevskii equation, in
fairly complete detail compared to Appendix 11.A of the preprint in the previous Chapter.
We find that this equation of motion can be written as a sum of terms multiplied by powers
of , corresponding to the case for a linear Schrödinger equation used for example by Zurek
and Paz [1] to investigate the semiclassical limit of quantum mechanics. One approaches
the equation of motion for the Wigner function describing Gross-Pitaevskii dynamics in a
similar way, and we can determine a corresponding semiclassical limit.
D.2
Definitions
We use Wigner’s original definition of the Wigner function for a pure state [2, 3, 4]
1
W (x, p) =
2π
Z
∞
−∞
dτ e−ipτ / ϕ∗ (x − τ /2)ϕ(x + τ /2).
(D.1)
To determine the desired equation of motion, we must first take the time derivative, to get
∞
∂ ∗
∂
∗
dτ e
ϕ (x − τ /2)ϕ(x + τ /2) + ϕ (x − τ /) ϕ(x + τ /2) ,
∂t
∂t
−∞
(D.2)
where the dynamics of ϕ(x) are governed by the Gross-Pitaevskii equation, written in general
form as
2
∂2
∂
2
+ V (x) + u|ϕ(x)| ϕ(x).
ϕ(x) = −
(D.3)
i
∂t
2m ∂x2
∂
1
W (x, p) =
∂t
2π
Z
−ipτ /
183
184
APPENDIX D. WIGNER FUNCTION FORMULATION. . .
We decide to consider the part of Eq. (D.2) dealing with the effects of the nonlinearity
separately from the rest. We thus rephrase Eq. (D.2):
∂
∂
∂
W (x, p) = W (x, p)SP + W (x, p)NL ,
∂t
∂t
∂t
(D.4)
where we have split up the differential equation into a part which is governed by the single
particle linear dynamics (SP), by which we mean the equation is the same as that for a single
quantum mechanical particle, and a part which is governed by the nonlinearity (NL).
D.3
Single-particle dynamics
We first consider the effective single particle dynamics, which are described by:
i
∂
W (x, p)SP =
∂t
2π
2
Z
∞
dτ e
−iτ p/
−∞
h
hϕ|Ĥ|x − τ /2ihx + τ /2|ϕi
i
−hϕ|x − τ /2ihx + τ /2|Ĥ|ϕi . (D.5)
This is exactly equivalent to the dynamics described by a linear Schrödinger equation.
Using Fourier’s integral theorem [5] and the Campbell-Baker-Hausdorff relation [6], we
find that
Z ∞
Z ∞Z ∞
i
∂
1
0
W (x, p)SP =
dτ
dφdx e−i(φx+τ p)/
2
∂t
2π
2π −∞ −∞
−∞
h
i 0
0
0
0
iτ p̂/2
iτ p̂/2 iφx̂/
iτ p̂/2
iτ p̂/2 iφx̂/
×hϕ| Ĥe
e
|x ihx |e
−e
e
|x ihx |e
Ĥ |ϕi
Z ∞Z ∞
i o
h
n
i
iτ p̂/2 iφx̂/ iτ p̂/2
−i(φx+τ p)/
|ϕi
hϕ|
e
e
Ĥ,
e
dφdτ
e
=
(2π )2 −∞ −∞
Z ∞Z ∞
h
i o
n
i
i(φx̂+τ p̂)/
−i(φx+τ p)/
=
hϕ|
Ĥ,
e
|ϕi .
(D.6)
dφdτ
e
(2π )2 −∞ −∞
We now set the commutator to be
i
h
i(φx̂+τ p̂)/
= i Ĉ,
Ĥ, e
(D.7)
where Ĉ is some yet to be determined operator. We know that [2, 7]
Z ∞Z ∞
hϕ|Ĉ|ϕi =
dxdpC(x, p)W (x, p),
−∞
(D.8)
−∞
where C(x, p) is a phase function, defined such that if
Z ∞Z ∞
1
Ĉ =
dφdτ c(φ, τ )ei(φx̂+τ p̂)/
2π −∞ −∞
(D.9)
185
D.3. SINGLE-PARTICLE DYNAMICS
then
We also know that
1
C(x, p) =
2π
Z
∞
−∞
Z
∞
dφdτ c(φ, τ )ei(φx+τ p)/ .
(D.10)
−∞
i C = {H, ei(φx+τ p)/ }MB ,
(D.11)
where the MB stands for Moyal bracket [2, 7], and H is the classical single particle Hamiltonian function. The Moyal bracket, as well as the identities above to do with the phase
function C, are derived in Appendix G.
From this we get
"
2s+1
∞
2 X (−1)s
C=
(2s + 1)! 2
s=0
#
2s+1
X (−1)t (2s + 1)! ∂ 2s+1−t ∂ t ∂ 2s+1−t ∂ t
. (D.12)
H
ei(φx+τ p)/
×
2s+1−t ∂pt
2s+1−t ∂xt
(2s
+
1
−
t)!t!
∂x
∂p
t=0
Because the p dependence of H is simply quadratic, the only possible values of t are 0, where
s may be any value, and 1, where s = 0 only. Thus,
(∞ "
#
)
2s+1 2s+1
∂
∂ 2s+1 i(φx+τ p)/
∂ i(φx+τ p)/
∂
2 X (−1)s
H
e
H e
C=
−
.
(2s + 1)! 2
∂x2s+1 ∂p2s+1
2 ∂p ∂x
s=0
(D.13)
i(φx̂+τ p̂)/
We therefore have an expression for hϕ|[Ĥ, e
]|ϕi, which we can substitute into
Eq. (D.6), resulting in
Z ∞Z ∞
Z ∞Z ∞
i
∂
0
0
−i(φx+τ p)/
i
dx dp
W (x, p)SP =
dφdτ e
2
∂t
(2π )
−∞ −∞
( " ∞ −∞ −∞ 2s+1
∂ 2s+1
∂ 2s+1 i(φx0 +τ p0 )/
2 X (−1)s
0
0
H(x
,
p
)
e
×
(2s + 1)! 2
∂x2s+1
∂p0 2s+1
s=0
#
)
0
0
∂
∂ i(φx +τ p )/
0
0
0
0
−
W (x , p ) . (D.14)
0 H(x , p )
0e
2 ∂p
∂x
Using the chain rule, and assuming all derivatives of W (x, p) = 0 at x, p = ±∞, this can be
rearranged to give
Z ∞Z ∞
Z ∞Z ∞
∂
2
0
0 i(φx0 +τ p0 )/
−i(φx+τ p)/
dx
dp
e
dφdτ
e
W (x, p)SP = −
∂t
(2π )2 −∞ −∞
−∞ −∞
(∞ X (−1)s (− /2)2s+1 ∂ 2s+1
∂ 2s+1
0
0
0
0
H(x , p ) 0 2s+1 W (x , p )
×
(2s + 1)!
∂x2s+1
∂p
s=0
)
∂
∂
0
0
0
0
H(x , p ) 0 W (x , p ) . (D.15)
+
2 ∂p0
∂x
186
APPENDIX D. WIGNER FUNCTION FORMULATION. . .
It is now a simple matter to make use of Fourier’s integral theorem [5], and arrive at the
following formulation [4]:
2s 2s+1
∞
X
(−1)s
∂ 2s+1
∂
∂
∂
∂
W (x, p)SP =
H
W − H W.
(D.16)
2s+1
2s+1
∂t
(2s
+
1)!
2
∂x
∂p
∂p
∂x
s=0
This is an exact expression, describing the motion of a single noninteracting particle. In the
next section we determine a similar expression which accounts for the nonlinearity in the
Gross-Pitaevskii equation.
D.4
Nonlinear Dynamics
For a simple cubic nonlinearity u|ϕ|2ϕ,
Z ∞
∗
iu
∂
2
2
−iτ p/
W (x, p)NL =
|ϕ(x
−
τ
/2)|
−
|ϕ(x
+
τ
/2)|
ϕ (x − τ /2)ϕ(x + τ /2).
dτ
e
∂t
2π 2 −∞
(D.17)
This can be re-expressed as
Z ∞
Z ∞
h
iu
∂
0
0
−iτ p/
dτ e
dp W (x − τ /2, p )
W (x, p)NL =
2
∂t
2π
−∞
−∞
i Z ∞ 00
00
0
00
iτ p /
dp e
−W (x + τ /2, p )
W (x, p ) . (D.18)
−∞
0
0
One can take W (x − τ /2, p ) − W (x + τ /2, p ), and expand it as a McLaurin series, finally
arriving at:
0
0
W (x − τ /2, p ) − W (x + τ /2, p ) = −2
∞
X
(τ /2)2s+1 ∂ 2s+1
0
W (x, p ).
2s+1
(2s + 1)! ∂x
s=0
(D.19)
Substituting this back into Eq. (D.18), we get
Z
∞
2iu X (1/2)2s+1 ∞ 0 ∂ 2s+1
∂
0
W (x, p)NL = −
dp 2s+1 W (x, p )
2
∂t
2π s=0 (2s + 1)! −∞
∂x
Z ∞
Z ∞
00
00
00
−iτ p/
×
dτ e
dp τ 2s+1 eiτ p / W (x, p ). (D.20)
−∞
We note that
τ
2s+1 iτ p00 /
e
=
2s+1
i
−∞
∂ 2s+1 iτ p00 /
,
e
∂p00 2s+1
(D.21)
and use the chain rule and Fourier’s integral theorem [5] to determine the final result:
2s+1
Z ∞
∞
∂
∂
2iu X (− /2i)2s+1 ∂ 2s+1
0
0
W (x, p)NL = −
W (x, p). (D.22)
dp W (x, p )
2s+1
∂t
(2s + 1)! ∂x
∂p2s+1
−∞
s=0
187
D.5. COMBINED RESULT
D.5
Combined Result
Combining Eqs. (D.16) and (D.22), we get the Wigner function dynamics to all orders in
corresponding to the wavefunction dynamics governed by Gross-Pitaevskii equation with
arbitrary potential, in one dimension:
∞
X (−1)s
∂
W =
∂t
(2s + 1)!
s=0
2s
2
∂ 2s+1
∂ 2s+1
∂
∂
[H
+
uρ]
W − H W.
2s+1
2s+1
∂x
∂p
∂p ∂x
(D.23)
R∞
We have introduced the density ρ = −∞ dpW (x, p), for brevity, which acts in some ways
like a time dependent effective potential.
In an analogous manner to the single particle linear Schrödinger equation case, we can
determine a semiclassical limit, where → 0. The result is a Liouville-like equation:
∂
∂
∂
∂
∂
W =
[H + uρ] W − [H + uρ] W,
∂t
∂x
∂p
∂p
∂x
(D.24)
This derivation can be easily generalized for other nonlinearities and to two and three dimensions.
References
[1] W. H. Zurek and J. P. Paz, Phys. Rev. Lett. 72, 2508 (1994); W. H. Zurek and J. P. Paz,
Nuov. Cim. B 110, 611 (1995).
[2] L. E. Reichl, The Transition to Chaos In Conservative Classical Systems: Quantum
Manifestations (Springer-Verlag, New York 1992).
[3] C. W. Gardiner, Quantum Noise (Springer-Verlag, Berlin 1991)
[4] E. P. Wigner, Phys. Rev. 40, 749 (1932).
[5] M. R. Spiegel, Theory and Problems of Advanced Calculus (McGraw-Hill, New York
1988).
[6] A. Galindo and P. Pascual, Quantum Mechanics (Springer-Verlag, Berlin 1991).
[7] J. E. Moyal, Proc. Cambridge Phil. Soc. 45, 99 (1949).