A Hybrid Lagrangian Variation Method for Bose-Einstein Condensates in Optical Lattices

0.21 MB·English

by  Mark Edwards

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0 A Hybrid Lagrangian Variational Method for
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2 Bose–Einstein Condensates in Optical Lattices

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Mark Edwards1§, Lisa M. DeBeer1 Mads Demenikov1 Jacob
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1 Galbreath1 T. Joseph Mahaney1 Bryan Nelsen1 and Charles

W. Clark2
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e Department of Physics, Georgia Southern University, Statesboro, GA 30460–8031,
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USA
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o 2 Electron and Optical Physics Division, National Institute of Standards and
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t Technology, Gaithersburg, MD 20899, USA
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- Abstract. Solving the Gross–Pitaevskii (GP) equation describing a Bose–Einstein
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condensate (BEC) immersed in an optical lattice potential can be a numerically
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o demanding task. We present a variational technique for providing fast, accurate
c solutions of the GP equation for systems where the external potential exhibits rapid
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varation along one spatial direction. Examples of such systems include a BEC

1 subjected to a one–dimensional optical lattice or a Bragg pulse. This variational
v
method is a hybrid form of the Lagrangian Variational Method for the GP equation
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in which a hybrid trial wavefunction assumes a gaussian form in two coordinates
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3 while being totally unspecified in the third coordinate. The resulting equations of
1 motionconsistofaquasi–one–dimensionalGPequationcoupledtoordinarydifferential
0
equationsforthewidthsofthetransversegaussians. Weusethismethodtoinvestigate
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0 how an optical lattice can be used to move a condensate non–adiabatically.
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PACS numbers: 3.75.Fi, 67.40.Db, 67.90.+Z
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c Submitted to: J. Phys. B: At. Mol. Phys.
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§ Corresponding author ([email protected])
Hybrid Lagrangian Variational Method 2
1. Introduction
Over the past several years, intense experimental and theoretical interest has focused
on the interaction of laser light with gaseous Bose–Einstein condensates (BEC’s)[1].
Examples include Bragg pulses applied to condensates and condensates confined in
optical lattices[2]. Experimental and theoretical studies in these areas are important
because condensates can be manipulated with Bragg pulses and strong optical lattices
can confine a definite number of condensate atoms in each well. The Mott–insulator
phasetransition[3]canproduceasystem that might becomeacandidateforprototypical
quantum computer.
This paper describes a method for rapidly finding accurate solutions of the Gross–
Pitaevskii equation describing a BEC in the presence of laser light. Such cases provide
some of the most demanding numerical computations in the field of gaseous Bose–
Einstein condensation. The “Hybrid” Lagrangian Method (HLM) described here is
an instance of the Lagrangian Variational Method (LVM) [4], a method for finding
approximate solutions of the Gross–Pitaevskii (GP) equation, when the condensate
wavefunction exhibits rapid variation along a single spatial direction. Examples of this
include Bose–Einstein condensates inopticallattices andBraggandstanding–wave laser
pulses to condensates.
The method presented here is similar to several other methods presented in the
literature[5]–[9]. The present method does have some significant differences. The most
important difference is that the motion of the condensate transverse to the direction
of the laser–light is coupled to the condensate motion along the light direction and
vice versa. This coupling presents a challenge to find the correct, self–consistent, and
variationally appropriate initial conditions. Such self–consistent initial conditions are
not necessary in the other methods cited.
This paper is organized as follows. The equations describing the basic Lagrangian
Variational Method are derived by writing down the trial wavefunction and describing
the meanings ofthe parameters. The steps required to arrive atthe finalHLM equations
ofmotionarepresented. ThemethodisthenappliedtothecaseofaBECinthepresence
of an accelerated optical lattice to show that it is possible to use a lattice to move a
condensate to a different location in a non–adiabatic way.
2. The Lagrangian Variational Method
The standard LVM is a procedure for obtaining approximate solutions of the time–
dependent Gross–Pitaevskii (GP) equation when a trial solution containing unknown
parameters is assumed. This procedure can be summarized as follows. First, a
Lagrangian density whose Euler–Lagrange equation is the GP equation is written down.
Next, a trial wavefunction containing several time–dependent variational parameters is
chosen. The Lagrangian whose generalized coordinates are these parameters is then
obtainedbyinserting thetrialwavefunction intotheLagrangiandensity andintegrating.
Hybrid Lagrangian Variational Method 3
The time–evolution equations for these parameters are derived using the usual Euler–
Lagrange equations for the new generalized coordinates.
It ispossibleto assume noparticularformforthetrialwavefunction andwritedown
anEuler–Lagrange–type equationinvolving theLagrangiandensity thatyields theexact
Gross–Pitaevskii partial differential equation. If one does assume a particular functional
form for the trial wavefunction having time–dependent variational parameters, then
equations of motion for these parameters are derived by integrating the Lagrangian
density over allspaceyielding anordinaryLagrangiandepending onlyontheparameters
and then applying standard Euler–Lagrange equations.
The “Hybrid” Variational Method combines these two ideas in that the trial
wavefunction assumed leaves the part of the wavefunction that depends on a particular
coordinate completely unspecified while assuming a particular functional form (with
time–dependent variational parameters) for the part of the wavefunction depending on
the other coordinates. The resulting equation of motion for the unspecified part of the
wavefunction isa partialdifferential equation while theequations ofmotionfortheother
variational parameters are ordinary differential equation that are first–order in time.
The time–dependent GP equation for a confined condensate that is subjected to
laser light has the following form.
∂ψ h¯2
ih¯ = − ∇2ψ +V (r,t)ψ(r,t)+V (r,t)ψ(r,t)+g|ψ(r,t)|2ψ(r,t). (1)
trap laser
∂t 2m
The Lagrangian density,
1 ∂ψ∗ ∂ψ h¯2
L(ψ∗,∂ ψ∗,∂ ψ∗,∂ ψ∗,∂ ψ∗;r,t) ≡ ih¯ ψ −ψ∗ + |∇ψ|2
x y z t
2 ∂t ∂t ! 2m
1
+ (V (r,t)+V (r,t))|ψ|2 + g|ψ|4 (2)
trap laser
2
yields the GP equation when the variation
δL ∂ ∂L ∂ ∂L ∂ ∂L ∂ ∂L ∂L
≡ + + + − (3)
δψ∗ ∂t ∂(∂tψ∗)! ∂x ∂(∂xψ∗)! ∂y ∂(∂yψ∗)! ∂z ∂(∂zψ∗)! ∂ψ∗
vanishes. This is the Euler–Lagrange equation that applies when the wavefunction is
completely unspecified.
To obtain a variationally approximate solution to the GP equation, we assume a
trial wavefunction that contains a set of variational parameters. Each parameter is
assumed to be a function of time. The most common example of a trial wavefunction
is a three-dimensional gaussian: [4]
ψ(A(t),w(t),b(t);r) = A(t)e−x2/wx2(t)+iβx(t)x2e−y2/wy2(t)+iβy(t)y2e−z2/wz2(t)+iβz(t)z2 (4)
where w ≡ (w ,w ,w ) and b ≡ (β ,β ,β ).
x y z x y z
By inserting this trial wavefunction into Eq. (3) and integrating over all space we
obtain the ordinary Lagrangian:
L(A(t),w(t),b(t)) = L[ψ(A(t),w(t),b(t);r)] d3r. (5)
Z
Hybrid Lagrangian Variational Method 4
In this way, we obtain a Lagrangian in which the variational parameters are the
generalized coordinates. Equations for the time dependence of these coordinates
are obtained from the usual Euler–Lagrange equations. For example, for the width
parameter w we would have the following.
x
d ∂L ∂L
− = 0. (6)
dt ∂w˙x! ∂wx
If one carries out this procedure for the gaussian trial wavefunction given above one
finds, after a little algebra, the following equation of motion for w
x
h¯2 π ah¯2N
w¨ +ω2w = + (7)
x x x m2w3 2 m2w2w w
x r x y z
Thus one obtains either a partial differential equation if the Euler–Lagrange equation
for the Lagrangian density is used or an ordinary differential equation for a variational
parameter if the integrated Lagrangian is used. The HVM combines these ideas and an
example of the HVM will be derived next.
3. The HVM equations of motion
If a BEC is immersed in a one–dimensional optical lattice or subjected to optical
laser pulses, then it is often the case that the condensate wavefunction exhibits spatial
oscillations along the direction of propagation of the incident laser light that are much
more rapid than along the transverse directions. In this case we can write a trial
wavefunction with the following form:
ψ(φ(x,t),w (t),w (t),β (t),β (t)) = φ(x,t)e−y2/2wy2(t)+iβy(t)y2e−z2/2wz2(t)+iβz(t)z2 (8)
y z y z
Notice here that the variational parameters are φ, w , w , β , and β . We have chosen
y z y z
to represent the shape of the wavefunction in the transverse direction with a gaussian
profile. InmanycasestheThomas-Fermiapproximationholdsandthetransversedensity
is more accurately represented by an inverted parabola. We shall see in section 3.3.2
that the size of the variationally determined density agrees well with the Thomas-Fermi
approximation when that limit applies.
We shall assume that this trial wavefunction is normalized like this
d3r|ψ|2 = N, (9)
Z
where N is the number of condensate atoms. The normalization condition constrains
the values of the variational parameters
+∞ +∞ +∞
d3r|ψ|2 = dx|φ(x,t)|2 dye−y2/wy2 dze−z2/wz2
−∞ −∞ −∞
Z Z Z Z
+∞
= π1/2w π1/2w dx|φ(x,t)|2 = N. (10)
y z
−∞
(cid:16) (cid:17)(cid:16) (cid:17)Z
Thus the wavefunction for the fast direction, φ(x,t), must satisfy
+∞ N
dx|φ(x,t)|2 = . (11)
−∞ πw w
Z y z
This can often be used to eliminate some variational parameters from the Lagrangian.
Hybrid Lagrangian Variational Method 5
3.1. The Hybrid Lagrangian
Since the functional dependence of the trial wavefunction on the y and z coordinates
is assumed to be gaussian, we shall treat them in the standard LVM way. That is, we
shall integrate the Lagrangian density only over these coordinates to get a “hybrid”
Lagrangian
+∞ +∞
∗ ∗ ∗
L(φ ,∂ φ ,∂ φ ,w ,w ,β ,β ) = dy dzL[ψ(r,t)]. (12)
x t y z y z
−∞ −∞
Z Z
where we take the full Lagrangian density to be
1 ∂ψ∗ ∂ψ h¯2
L(ψ∗,∂ ψ∗,∂ ψ∗,∂ ψ∗,∂ ψ∗;r,t) ≡ ih¯ ψ −ψ∗ + |∇ψ|2
x y z t
2 ∂t ∂t ! 2m
1
+V (r,t)|ψ|2 + g|ψ|4 (13)
ext
2
and where
V (r,t) = V (r,t)+V (x,t). (14)
ext trap laser
We willassume that theconfining potential, V , has the formof anharmonic oscillator
trap
and that the potential created by the light field, V , depends only on the x coordinate.
laser
The explicit forms of these are
1 1 1
V (r,t) = mω2x2 + mω2y2 + mω2z2
trap 2 x 2 y 2 z
Ω2(t)
V (x,t) = −h¯ 0 (1+cos(2k x−δ t)) (15)
laser L L
2∆
where m is the atomic mass, (ω ,ω ,ω ) are the frequencies of the harmonic confining
x y z
potential, k is the wavevector of the laser and δ is the difference in frequencies of the
L L
two counterpropagating beams that comprise the laser pulse, Ω is the single–photon
0
Rabi frequency of the condensate atom for the laser pulse, and ∆ is the detuning of the
incident laser pulse from atomic resonance.
Calculating the form of the hybrid Lagrangian for the given trial wavefunction (Eq.
(8) consists of (1) inserting the wavefunction into the full Lagrangian density, and (2)
performing the integration over the y and z coordinates. We shall carry out these steps
for each separate term of the full Lagrangian
∗
1 ∂ψ ∂ψ ∂ψ
∗ ∗
L = ih¯ ψ −ψ = h¯Im ψ
1
2 ∂t ∂t ! ( ∂t )
h¯2 h¯2 ∂ψ∗∂ψ ∂ψ∗∂ψ ∂ψ∗ ∂ψ
L = |∇ψ|2 = + + +
2
2m 2m! ∂x ∂x ∂y ∂y ∂z ∂z !
1 1 1
L = V (r,t)|ψ|2 = mω2x2 + mω2y2 + mω2z2 +V (x,t) |ψ|2
3 ext 2 x 2 y 2 z laser
(cid:18) (cid:19)
1
L = g|ψ|4 (16)
4
2
Then the resulting form of the hybrid Lagrangian for the given trial wavefunction is
found from
+∞ +∞
∗ ∗ ∗
L (φ ,∂ φ ,∂ φ ,w,β) = dy dzL [ψ(r,t)]. i = 1,...,4 (17)
i x t i
−∞ −∞
Z Z
Hybrid Lagrangian Variational Method 6
where w = (w ,w ) and β = (β ,β ). The hybrid Lagrangian is found by inserting
x x x x
the trial wavefunction into the above and performing the required differentiations and
integrations.
The result is
h¯ ∂φ ∂φ∗ h¯2 ∂φ∗∂φ 1
L (φ∗,∂ φ∗,∂ φ∗,w,β) = φ∗ −φ + + h¯β˙ w2
hybrid x t (2i ∂t ∂t ! 2m! ∂x ∂x "2 y y
1 h¯2 1 1
+ h¯β˙ w2 + +2β2w2 + +2β2w2
2 z z 2m! 2w2 y y 2w2 z z!
y z
1 1 1
+ V (x,t)+ mω2x2 + mω2w2 + mω2w2 |φ|2
laser 2 x 4 y y 4 z z#
1
+ g|φ|4 π1/2w π1/2w (18)
y z
4 )
(cid:16) (cid:17)(cid:16) (cid:17)
We now present the derivation of the equations of motionfor the variationalparameters.
3.2. The Euler–Lagrange equations of motion
The next step is to derive the equations of motion for φ, w , w , β , and β . We shall
y z y z
consider each in turn. We shall then discover that some further algebra will be required
to make the resulting equations somewhat more convenient for calculation.
3.2.1. The equation of motion for φ(x,t) The Euler–Lagrange equation of motion for
∗
the variational parameter φ is given by
∂ ∂L ∂ ∂L ∂L
hybrid hybrid hybrid
+ − = 0. (19)
∂t ∂ ∂φ∗  ∂x ∂ ∂φ∗  ∂φ∗
∂t ∂x
 (cid:16) (cid:17)  (cid:16) (cid:17)
To evaluate this equation for our case we first compute the derivatives of L with
hybrid
∗ ∗ ∗
respect toφ , ∂ φ , and∂ φ . Inserting these derivatives intoEq. (19)gives thefollowing
t x
equation of motion for φ:
∂φ 1 w˙ w˙ h¯2 ∂2φ 1 1
ih¯ + ih¯ y + z φ = − + V (x,t)+ mω2x2 φ+ g|φ|2φ
∂t 2 wy wz! 2m∂x2 (cid:18) laser 2 x (cid:19) 2
1 h¯2 1 1
+ h¯β˙ w2 + +2β2w2 + mω2w2
"2 y y 2m 2w2 y y! 4 y y
y
1 h¯2 1 1
+ h¯β˙ w2 + +2β2w2 + mω2w2 φ (20)
2 z z 2m 2w2 z z! 4 z z#
z
By defining
1 h¯2 1 1
h¯f (t) ≡ h¯β˙ w2 + +2β2w2 + mω2w2, i = x,y (21)
i 2 i i 2m 2wi2 i i! 4 i i
and
1
U (x,t) ≡ V (x,t)+ mω2x2 (22)
ext laser 2 x
Hybrid Lagrangian Variational Method 7
we can write the equation of motion for φ as
∂φ 1 w˙ w˙ h¯2 ∂2φ 1
ih¯ +ih¯ y + z +i[f +f ] φ = − +U (x,t)φ+ g|φ|2φ (23)
∂t 2 "wy wz# x y ! 2m∂x2 ext 2
We can simplify this equation considerably using the following transformation:
φ(x,t) = N1/2φ˜(x,t)e−a(t)+ib(t) (24)
where
′ ′
1 t w˙ (t) w˙ (t) t
y z ′ ′ ′ ′
a(t) ≡ + dt and b(t) ≡ − (f (t)+f (t))dt (25)
2 Z0 "wy(t′) wz(t′)# Z0 y z
The result is
ih¯∂φ˜ = − h¯2 ∂2φ˜ +U (x,t)φ˜+ 1gNe−2a(t) φ˜2φ˜ (26)
∂t 2m∂x2 ext 2
(cid:12) (cid:12)
This equation can be simplified even further by noting that(cid:12) (cid:12)
(cid:12) (cid:12)
w (0)w (0)
e−2a(t) = y z . (27)
w (t)w (t)
y z
˜
The final form for the equation of motion for φ is the following
ih¯∂φ˜ = − h¯2 ∂2φ˜ + V (x,t)+ 1mω2x2 φ˜+ 1gN wy(0) wz(0) φ˜2φ˜. (28)
∂t 2m∂x2 (cid:18) laser 2 x (cid:19) 2 wy(t)! wz(t)!(cid:12) (cid:12)
(cid:12) (cid:12)
This is the equation of motion for φ˜. We now turn to the equations of m(cid:12)ot(cid:12)ion for the
widths and phases w and β .
i i
3.2.2. The equations of motion for the phases, β We turn now to the Euler–Lagrange
i
equations for the gaussian phases. These equations will provide a relationship between
the phases and the gaussian widths. These relationships will be useful in simplifying
the equations for the widths.
The Euler–Lagrange (EL) equations for the gaussian phases, β and β , in terms
y z
of the hybrid Lagrangian are as follows:
∂ ∂L ∂L
hybrid hybrid
− = 0, i = y,z. (29)
∂t ∂β˙i ! ∂βi
Computing the required derivatives and inserting them into the EL equation gives
∂ 1 h¯2
h¯w2|φ(x,t)|2 π1/2w π1/2w = 4β w2 |φ(x,t)|2π1/2w π1/2w (30)
∂t (cid:20)2 y (cid:16) y(cid:17)(cid:16) z(cid:17)(cid:21) "2m#(cid:16) y y(cid:17) y z
and
∂ 1 h¯2
h¯w2|φ(x,t)|2 π1/2w π1/2w = 4β w2 |φ(x,t)|2π1/2w π1/2w (31)
∂t (cid:20)2 z (cid:16) y(cid:17)(cid:16) z(cid:17)(cid:21) "2m#(cid:16) z z(cid:17) y z
We can simplify these equations by first considering the left–hand–side:
∂ 1 ∂ 1
h¯w2|φ(x,t)|2 π1/2w π1/2w = h¯w2 |φ(x,t)|2 π1/2w π1/2w
∂t 2 y y z ∂t 2 y y z
(cid:20) (cid:16) (cid:17)(cid:16) (cid:17)(cid:21) (cid:18) (cid:19) (cid:16) (cid:17)(cid:16) (cid:17)
1 ∂
+ h¯w2 |φ(x,t)|2 π1/2w π1/2w (32)
2 y∂t y z
h (cid:16) (cid:17)(cid:16) (cid:17)i
Hybrid Lagrangian Variational Method 8
Thus we can write the EL equation for β as
y
h¯2 ∂ 1
4β w2 |φ(x,t)|2π1/2w π1/2w = h¯w2 |φ(x,t)|2 π1/2w π1/2w
"2m#(cid:16) y y(cid:17) y z ∂t (cid:18)2 y(cid:19) (cid:16) y(cid:17)(cid:16) z(cid:17)
1 ∂
+ h¯w2 |φ(x,t)|2 π1/2w π1/2w (33)
2 y∂t y z
h (cid:16) (cid:17)(cid:16) (cid:17)i
To simplify this further, multiply both sides of the above equation by dx and integrate:
h¯2 +∞ ∂ 1
4β w2 dx |φ|2 π1/2w π1/2w = h¯w2
2m!(cid:16) y y(cid:17)Z−∞ (cid:16) y(cid:17)(cid:16) z(cid:17) ∂t (cid:18)2 y(cid:19)
+∞
× dx |φ|2 π1/2w π1/2w
y z
−∞
Z (cid:16) (cid:17)(cid:16) (cid:17)
1
+ h¯w2
2 y
+∞ ∂
× dx |φ|2 π1/2w π1/2w
y z
−∞ ∂t
Z h (cid:16) (cid:17)(cid:16) (cid:17)i
(34)
We assume that the derivative and the integral can be interchanged in the last term on
the right. Given this to be true, we can use the normalization condition (Eq. (10))
+∞
π1/2w π1/2w dx|φ(x,t)|2 = N (35)
y z
−∞
(cid:16) (cid:17)(cid:16) (cid:17)Z
to show that the last term in Eq. (34) is zero and thus we have
h¯2 ∂ 1
N 4β w2 = N h¯w2
2m!(cid:16) y y(cid:17) ∂t (cid:18)2 y(cid:19)
h¯ ∂
4β w2 = w2 = 2w w˙
m! y y ∂t y y y
(cid:16) (cid:17) (cid:16) (cid:17)
h¯
2β w = w˙ (36)
y y y
m!
The EL equation for β is similar and so we finally have
z
m w˙
y
β =
y
2h¯ w
(cid:18) (cid:19) y
m w˙
z
β = (37)
z
2h¯ w
(cid:18) (cid:19) z
˙
These equations can be used to eliminate β and β (i = x,y) in the expressions for the
i i
f .
i
1 h¯2 1 1
h¯f (t) = h¯β˙ w2 + +2β2w2 + mω2w2,
i 2 i i 2m 2wi2 i i! 4 i i
1 h¯2 1 1
= mw w¨ + + mω2w2, i = x,y. (38)
4 i i 2m2w2 4 i i
i
This will be useful in simplifying the equations for the gaussian widths to which we now
turn.
Hybrid Lagrangian Variational Method 9
3.2.3. The equations of motion for the widths, w We next consider the EL equations
i
for the widths w , where i = x,y. The EL equation for the w is given by
i i
∂ ∂L ∂L
hybrid hybrid
− = 0. (39)
∂t ∂w˙i ! ∂wi
We shall derive the equation for w ; the derivation of the equation for w is similar.
y z
Inspection shows that L is independent of w˙ and so the above equation becomes
hybrid y
∂L
hybrid
= 0. (40)
∂w
y
We can easily compute this derivative:
∂L ∂φ h¯2 ∂φ∗∂φ 1
hybrid = h¯Im φ∗ + +V (x,t)|φ|2 + mω2x2|φ|2
∂wy ( ( ∂t) 2m! ∂x ∂x laser 2 x
g 1 h¯2 1 1
+ |φ|4 + h¯β˙ w2|φ|2 + +2β2w2 |φ|2 + mω2w2|φ|2
4 2 y y 2m! 2w2 y y! 4 y y
y
1 h¯2 1 1
+ h¯β˙ w2|φ|2 + +2β2w2 |φ|2 + mω2w2|φ|2
2 z z 2m! 2w2 z z! 4 z z )
z
× π1/2 π1/2w
z
(cid:16) (cid:17)(cid:16) h¯2(cid:17) 1 1
+ h¯β˙ w + − +4β2w + mω2w
( y y 2m! w3 y y! 2 y y)
y
× π1/2w π1/2w |φ|2 (41)
y z
(cid:16) (cid:17)(cid:16) (cid:17)
Defining
∂φ h¯2 ∂φ∗∂φ 1 1
H ≡ h¯Im φ∗ + +V (x,t)|φ|2 + mω2x2|φ|2 + g|φ|4, (42)
x ( ∂t) 2m! ∂x ∂x laser 2 x 4
we can rewrite Eq. (40) as
∂L 3 h¯2 1 3
hybrid = h¯β˙ w + − +6β2w + mω2w
∂wy (2 y y 2m! 2wy3 y y! 4 y y
1 w2 h¯2 1 w2 1 w2
+ h¯β˙ z + +2β2 z + mω2 z
2 zwy 2m! 2wywz2 zwy! 4 zwy)
H
× π1/2w π1/2w |φ|2 + x π1/2w π1/2w
y z y z
w
y
(cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17)(cid:16) (cid:17)
= 0. (43)
By multiplying by dx on both sides and integrating over x we obtain
∂L 3 h¯2 1 3
hybrid = N h¯β˙ w + − +6β2w + mω2w
∂wy (2 y y 2m! 2wy3 y y! 4 y y
1 w2 h¯2 1 w2 1 w2
+ h¯β˙ z + +2β2 z + mω2 z
2 zwy 2m! 2wywz2 zwy! 4 zwy)
+ πhH iw
x z
= 0. (44)
Hybrid Lagrangian Variational Method 10
where
+∞ ∂φ h¯2 ∂φ∗∂φ
hH i = dx h¯Im φ∗ + +V (x,t)|φ|2
x laser
−∞ " ( ∂t) 2m! ∂x ∂x
Z
1 g
+ mω2x2|φ|2 + |φ|4 (45)
2 x 4 #
By using the equations of motion for the phases and their time derivatives we can
˙
eliminate the β and β from Eq. (44)
i i
∂L 3 1 w h¯2 1 1
hybrid z
= N mw¨ + m w¨ + − +
∂wy (4 y 4 wy z 2m! 2wy3 2wywz2!
3 1 w2
+ mω2w + mω2 z +πhH iw = 0. (46)
4 y y 4 zwy) x z
The equation for w is obtained by writing the above equation down with y and z
z
interchanged. To simplify these equations further we must consider the expression for
hH i.
x
˜
We can express hH i in terms of φ, defined in Eq. (24), by inserting that equation
x
into Eq. (45) to get
hH i = Ne−2a(t)hH˜ i+Ne−2a(t)h¯b˙(t) +∞dx φ˜2 (47)
x x
−∞
Z (cid:12) (cid:12)
where (cid:12) (cid:12)
(cid:12) (cid:12)
+∞ ∂φ˜ h¯2 ∂φ˜∗∂φ˜
hH˜ i ≡ dx h¯Im φ˜∗ +
x
−∞ ( ( ∂t) 2m! ∂x ∂x
Z
+ V (x,t)+ 1mω2x2 φ˜2 + 1gNe−2a(t) φ˜4 . (48)
laser 2 x 4 )
(cid:18) (cid:19)(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
This can be significantly simplified by usin(cid:12)g(cid:12)the equation of(cid:12)m(cid:12)otion for φ˜.
hH˜ i = − +∞dx 1gNe−2a(t) φ˜4 (49)
x
−∞ 4
Z (cid:12) (cid:12)
Thus we can write a compact expression for(cid:12)h(cid:12)H i. Using Eq. (47) we have
(cid:12) (cid:12) x
hH i = Ne−2a(t) − +∞dx 1gNe−2a(t) φ˜4 −h¯(f (t)+f (t)) +∞dx φ˜2 (50)
x y z
−∞ 4 −∞
(cid:18) Z (cid:12) (cid:12) Z (cid:12) (cid:12) (cid:19)
Using the above expression we are(cid:12) a(cid:12)t last ready to build the fin(cid:12)al(cid:12)equations of
(cid:12) (cid:12) (cid:12) (cid:12)
motion for the width parameters. The results are
h¯2 1 2λ
w¨ +ω2w = + (51)
y y y m2! wy3 wy2wz
h¯2 1 2λ
w¨ +ω2w = + , (52)
z z z m2! wz3 wywz2
where
λ φ˜ ≡ πw2(0)w2(0) +∞dx 1gN φ˜4. (53)
y z −∞ 4 m
(cid:16) (cid:17) Z (cid:12) (cid:12)
This completes the full set of equations of motion f(cid:12)or(cid:12) the gaussian parameters.
(cid:12) (cid:12)