2.1 The potential step

d dy snc�<sup>1</sup>

Nonlinear Optics ‐ Novel Results in Theory and Applications

d dy ncc�<sup>1</sup>

d dy nsc�<sup>1</sup>

h

dnc<sup>00</sup>

ncc00ð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>n</sup>6nccð Þ� <sup>u</sup>; <sup>α</sup> <sup>2</sup>αcnc<sup>∗</sup>

ndc<sup>00</sup>

tac00ð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>1</sup> <sup>þ</sup> tac2 ½ � ð Þ <sup>u</sup>; <sup>α</sup>

Now, the second derivatives are as follows

ð Þ¼� y

ð Þ¼� y

ð Þ¼� y

ð Þ¼ <sup>u</sup>; <sup>α</sup> 2 dnc<sup>2</sup>

ð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>2</sup>α<sup>2</sup> ndc<sup>3</sup>

associated with cnc ð Þ u; α and snc ð Þ u; α are given by

j

j s

38

coc00ð Þ¼ <sup>u</sup>; <sup>α</sup> 2 1 � coc<sup>2</sup> ½ � ð Þ <sup>u</sup>; <sup>α</sup> cocð Þ <sup>u</sup>; <sup>α</sup> dnc<sup>2</sup>

� <sup>2</sup>αab cnc<sup>∗</sup>ð Þ <sup>u</sup>; <sup>α</sup>

<sup>c</sup>ð Þ¼ <sup>u</sup> <sup>R</sup><sup>e</sup> cnc<sup>∗</sup>ð Þ� <sup>u</sup>; <sup>α</sup> <sup>i</sup> <sup>d</sup>

ð Þ¼ <sup>u</sup> <sup>R</sup><sup>e</sup> snc<sup>∗</sup>ð Þ� <sup>u</sup>; <sup>α</sup> <sup>i</sup> <sup>d</sup>

<sup>¼</sup> <sup>a</sup><sup>2</sup> � <sup>b</sup><sup>2</sup> � � dncð Þ <sup>u</sup>; <sup>α</sup> ,

<sup>¼</sup> <sup>a</sup><sup>2</sup> � <sup>b</sup><sup>2</sup> � � dncð Þ <sup>u</sup>; <sup>α</sup> ,

respectively. The nonlinear term causes that the quantum flux be no longer constant (as is the case for linear interaction) but modulated by dnc ð Þ u; α instead.

y

y

cnc00ð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>2</sup><sup>α</sup> j j cncð Þ <sup>u</sup>; <sup>α</sup> <sup>2</sup> � <sup>n</sup><sup>6</sup> cncð Þ� <sup>u</sup>; <sup>α</sup> <sup>2</sup>αab cnc<sup>∗</sup>ð Þ <sup>u</sup>; <sup>α</sup> ,

snc00ð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>3</sup><sup>α</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � � � <sup>1</sup> � � � <sup>2</sup>αj j sncð Þ <sup>u</sup>; <sup>α</sup> <sup>2</sup> � �

sncð Þ� <sup>u</sup>; <sup>α</sup> <sup>2</sup>αab snc<sup>∗</sup>

nsc00ð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>n</sup><sup>3</sup> <sup>þ</sup> <sup>8</sup>abn1nsc<sup>2</sup> ð Þ ð Þ <sup>u</sup>; <sup>α</sup> nscð Þ <sup>u</sup>; <sup>α</sup>

ð Þ� u; α αn<sup>0</sup>

<sup>α</sup>½ � 2 tacð Þþ <sup>u</sup>; <sup>α</sup> <sup>i</sup>ℑtacðu; <sup>α</sup><sup>Þ</sup> j j cncð Þ <sup>u</sup>; <sup>α</sup> <sup>2</sup> � 2 tacð Þ <sup>u</sup>; <sup>α</sup> n o

The first three of the above equations can be thought of as modifications of the Gross-Pitaevskii equation, which allows for solutions of the form cnc ð Þ u; α , snc ð Þ u; α , and dnc ð Þ u; α . However, when a or b vanishes, we get the Gross-Pitaevskii form. With these results at hand, we can see that the probability current densities

cncð Þ <sup>u</sup>; <sup>α</sup> � snc<sup>∗</sup>ð Þ <sup>u</sup>; <sup>α</sup>

� � � �

� � � �

� �

ð Þþ <sup>u</sup>; <sup>α</sup> <sup>2</sup>αabncc2

i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ab þ y<sup>2</sup> ð Þ n<sup>1</sup> þ αj j y

i

i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4aby<sup>2</sup> ð Þ n<sup>1</sup> þ α=j j y

ð Þ u; α ,

� <sup>α</sup> <sup>1</sup> <sup>þ</sup> <sup>10</sup>abnsc2 ½ � ð Þ <sup>u</sup>; <sup>α</sup> snc<sup>∗</sup>ð Þ <sup>u</sup>; <sup>α</sup> , (33)

<sup>þ</sup> <sup>2</sup><sup>α</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �ndcð Þ <sup>u</sup>; <sup>α</sup> , (34)

ð Þ u; α

ð Þ <sup>u</sup>; <sup>α</sup> ð Þ <sup>ℑ</sup>cncð Þ <sup>u</sup>; <sup>α</sup> sncðu; <sup>α</sup><sup>Þ</sup> <sup>2</sup>

sncð Þ u; α

du cncð Þ <sup>u</sup>; <sup>α</sup>

du sncð Þ <sup>u</sup>; <sup>α</sup>

� �dncð Þ <sup>u</sup>; <sup>α</sup> , (31)

ð Þ <sup>u</sup>; <sup>α</sup> cnc<sup>∗</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � 4aby<sup>2</sup> ð Þ 1 � α=j j y

2 <sup>r</sup> � � , (26)

2 <sup>r</sup> � � , (27)

2 <sup>r</sup> � � : (28)

(29)

(30)

ð Þ u; α , (32)

, (35)

(37)

(38)

cocð Þ <sup>u</sup>; <sup>α</sup> : (36)

A straight forward application of the functions introduced in this section is the finding of the eigenfunctions of the Gross-Pitaevskii equation for a step potential:

$$V(u) = \begin{cases} 0, & \text{when } u < 0, \\ V\_{0}, & \text{when } u \ge 0, \end{cases} \tag{39}$$

and a chemical potential μ larger than the potential height V0. The Gross-Pitaevskii equation is written as

$$\frac{d^2\psi(u)}{du^2} + \frac{2ML^2}{\hbar^2}(\mu - V\_0) \left|\psi(u) - \frac{2ML^2}{\hbar^2 A^2} N U\_0 |\psi(u)|^2 \right| \psi(u) = 0,\tag{40}$$

where ψð Þ u is the unnormalized eigenfunction for the Bose-Einstein condensate (BEC), M is the mass of a single atom, N is the number of atoms in the condensate, <sup>U</sup><sup>0</sup> <sup>¼</sup> <sup>4</sup>πћ<sup>2</sup> a=M characterizes the atom-atom interaction, a is the scattering length, L is a scaling length, A is the integral of the magnitude squared of the wave function, u is a dimensionless length, μ is the chemical potential, and V<sup>0</sup> is an external constant potential.

For u < 0 (we call it the region I, V<sup>0</sup> ¼ 0), we use the cnc function with a ¼ 1, i.e.,

$$\psi\_I(\mu, a) = \texttt{cnc}(k\_I \mu, a\_I), \tag{41}$$

with parameters

$$k\_I^2 = \frac{2ML^2\mu}{\hbar^2 \left[1 + a\_I \left(a^2 + b^2\right)\right]},\tag{42}$$

$$a\_{l} = \frac{ML^{2}NU\_{0}}{\hbar^{2}A^{2}k\_{I}^{2}}\tag{43}$$

From these equations, we obtain

$$a\_l = \frac{NU\_0}{2\mu A^2 - NU\_0 \left(a^2 + b^2\right)},\tag{44}$$

and

$$
\mu = \frac{\hbar^2 k\_I^2}{2ML^2} + \frac{NU\_0}{2A^2} (a^2 + b^2). \tag{45}
$$

This last result for μ is in agreement with the conjecture formulated by D'Agosta et al. in Ref. [9], with the last term being the self-energy of the condensate, which is independent of kI.

For u>0, we use the nonlinear plane wave (a ¼ T, b ¼ 0)

$$\psi\_{\rm II}(u) = \texttt{cnc}(k\_{\rm II}u, a\_{\rm II}),\tag{46}$$

with

$$\mathbf{1} + a\_{\mathrm{II}} \mathbf{T}^2 = \frac{2\mathbf{M}\mathbf{L}^2}{\hbar^2 k\_{\mathrm{II}}^2} (\mu - V\_0), \quad \frac{2\mathbf{M}\mathbf{L}^2 \mathbf{N} U\_0}{\hbar^2 \mathbf{A}^2 k\_{\mathrm{II}}^2} = 2a\_{\mathrm{II}},\tag{47}$$

i.e.,

$$\mu = V\_0 + \frac{\hbar^2 k\_{\rm II}^2}{2\text{ML}^2} + \frac{\text{NU}\_0}{2\text{A}^2} T^2, \quad k\_{\rm II}^2 = \frac{2\text{ML}^2 (\mu - V\_0)}{\hbar^2 \left(1 + a\_{\rm II} T^2\right)}. \tag{48}$$

By combining the expressions for the αs in both regions, we find that

$$
\alpha\_l \mathbb{k}\_I^2 = \alpha\_{l\mathbb{I}} \mathbb{k}\_{\text{II}}^2 \tag{49}
$$

3. Nonlinear superposition of trigonometric functions

Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential

<sup>1</sup> <sup>þ</sup> <sup>α</sup>

r

<sup>1</sup> <sup>þ</sup> <sup>α</sup>

snað Þ <sup>u</sup> , ocað Þ <sup>u</sup> <sup>≔</sup> <sup>1</sup>

snað Þ <sup>u</sup> , scað Þ <sup>u</sup> <sup>≔</sup> snað Þ <sup>u</sup>

cnað Þ <sup>u</sup> , sdað Þ <sup>u</sup> <sup>≔</sup> snað Þ <sup>u</sup>

The algebraic relationships between the above functions are

.

A plot of these functions can be found in Figure 4, for a set of values of α, a, b.

some results; more details are found in Ref. [10].

dθ du ¼

where <sup>α</sup>:a, b∈R, and <sup>∣</sup>α<sup>∣</sup> < 4∣ab∣<sup>=</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �<sup>2</sup>

Figure 3. Thus, the relationship between θ and u is

dnað Þ¼ u

DOI: http://dx.doi.org/10.5772/intechopen.80938

u ¼ ðθ 0

osað Þ <sup>u</sup> <sup>≔</sup> <sup>1</sup>

csað Þ <sup>u</sup> <sup>≔</sup> cnað Þ <sup>u</sup>

dcað Þ <sup>u</sup> <sup>≔</sup> dnað Þ <sup>u</sup>

Figure 3.

41

Three-dimensional plot of <sup>4</sup>∣ab∣<sup>=</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �<sup>2</sup>

We also define the nonlinear functions

A second set of nonlinear functions is the nonlinear version of the superposition of trigonometric functions, which is the subject of this section. We only mention

> dθ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>a</sup><sup>2</sup> � <sup>b</sup><sup>2</sup> � � cos 2ð Þþ <sup>θ</sup> <sup>α</sup>ab sin 2ð Þ<sup>θ</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>a</sup><sup>2</sup> � <sup>b</sup><sup>2</sup> � � cos 2ð Þþ <sup>θ</sup> <sup>α</sup>ab sin 2ð Þ<sup>θ</sup>

<sup>q</sup> : (56)

snað Þ u ≔a sin ð Þ� θ b cosð Þθ , (57) cnað Þ u ≔a cosð Þþ θ b sin ð Þθ , (58)

cnað Þ <sup>u</sup> , odað Þ <sup>u</sup> <sup>≔</sup> <sup>1</sup>

cnað Þ <sup>u</sup> , dsað Þ <sup>u</sup> <sup>≔</sup> dnað Þ <sup>u</sup>

dnað Þ <sup>u</sup> , cdað Þ <sup>u</sup> <sup>≔</sup> cnað Þ <sup>u</sup>

, a plot of 4∣ab∣<sup>=</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �<sup>2</sup>

, (55)

dnað Þ <sup>u</sup> , (59)

snað Þ <sup>u</sup> , (60)

dnað Þ <sup>u</sup> : (61)

, is shown in

Let us consider the change of variable from θ to u defined by the Jacobian

and since the chemical potential should be the same on both regions, we also get

$$V\_0 = \frac{\hbar^2 \left(k\_I^2 - k\_{II}^2\right)}{2\text{ML}^2} + \frac{\text{N}U\_0}{2\text{A}^2} \left(a^2 + b^2 - T^2\right). \tag{50}$$

The equal flux condition results in

$$k\_I \left(a^2 - b^2\right) = k\_{II} T^2. \tag{51}$$

Now, equating the functions and their derivatives at u ¼ 0, we find two relations for the parameters:

$$a + b = T,\tag{52}$$

$$k(a-b)k\_I\sqrt{1-a\_I(a+b)^2} = Tk\_{II}\sqrt{1-a\_{II}T^2},\tag{53}$$

i.e.,

$$\frac{k\_{\rm II}}{k\_{\rm I}} = \frac{(a-b)}{(a+b)} \sqrt{\frac{\mathbf{1} - a\boldsymbol{q}(\boldsymbol{a}+\boldsymbol{b})^2}{\mathbf{1} - a\boldsymbol{q}(\boldsymbol{a}+\boldsymbol{b})^2 k\_{\rm I}^2/k\_{\rm II}^2}}.\tag{54}$$

We show these values in Figure 2. We observe a behavior similar to the linear system; when μ≫V<sup>0</sup> (kII ! kI), which means very high energies, the step is just a small perturbation on the evolution of the wave.

Figure 2. A three-dimensional plot of the values of kII=kI for the potential step. Dimensionless units.

Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential DOI: http://dx.doi.org/10.5772/intechopen.80938
