Waves and Solitons

Abstract

In the previous chapter we considered the shape of steady state condensates, either homogeneous or confined by trapping potentials. We have seen that the condensate described by the GPE is a special kind of fluid, similar to the idealized Euler fluid without viscosity that appears in classical fluid dynamics textbooks. Not surprisingly for a fluid, the dynamics of the condensate exhibit a variety of interesting time-dependent phenomena, from sound waves and shape oscillations, to solitons and vortices.

Problems

4.1

For a dark soliton, the integrals of motion in Eqs. (4.35)–(4.37) are renormalized so as to remove the contribution from the background and lead to finite values,

$$\begin{aligned} N_\mathrm{s}= & {} \int \limits _{-\infty }^{+\infty }(n_0-|\psi |^2)~\mathrm{d}x \nonumber \\ \mathcal {P}_\mathrm{s}= & {} \frac{i\hbar }{2} \int \limits _{-\infty }^{+\infty }\left( \psi \frac{\partial \psi ^*}{\partial x} -\psi ^* \frac{\partial \psi }{\partial x}\right) \left( 1-\frac{n_0}{|\psi |^2} \right) \mathrm{d}x \nonumber \\ E_\mathrm{s}= & {} \int \limits _{-\infty }^{+\infty }\left( \frac{\hbar ^2}{2m}\left| \frac{\partial \psi }{\partial x} \right| ^2 + \frac{g}{2}(|\psi |^2-n_0)^2 \right) \mathrm{d}x \nonumber \end{aligned}$$

Evaluate these integrals using the dark soliton solution Eq. (4.38), leaving your answers in terms of $\xi , n_0, u$ and c.

4.2

Consider a dark soliton in a harmonically-trapped condensate. Approximating the background condensate with the Thomas–Fermi profile $n(x)=n_0(1-x^2/R_x^2)$ (for $x \le R_x$, otherwise $n=0$) and treating the soliton depth $n_\mathrm{d}$ to be constant, obtain an expression for the soliton speed as a function of its position x and depth $n_\mathrm{d}$. Hence obtain an expression for the turning points of its motion.

4.3

Show that the static ($u=0$) bright soliton solution, obtained from Eq. (4.52), is a solution to the 1D attractive time-independent GPE with $V(x)=0$, i.e.,

$$\begin{aligned} \mu \psi = -\frac{\hbar ^2}{2m}\frac{\mathrm{d}^2 \psi }{\mathrm{d}x^2}-|g||\psi |^2 \psi , \end{aligned}$$

(4.62)

and hence determine an expression for the chemical potential of the soliton.

4.4

Using the general bright soliton solution, Eq. (4.52), evaluate the soliton integrals of motion according to Eqs. (4.35)–(4.37). The soliton solution is already normalized to the number of atoms, N. Show that the bright soliton behaves as a classical particle with positive mass.

4.5

Consider a 3D bright soliton in a cylindrically-symmetric waveguide with tight harmonic confinement (of frequency $\omega _r$) in r and no trapping along z. We can construct the ansatz for the soliton,

$$\begin{aligned} \psi (z,r)=A \mathrm{sech} \left( \frac{z}{l_r \sigma _z} \right) \exp \left( -\frac{r^2}{2 l_r^2 \sigma _r^2} \right) , \end{aligned}$$

(4.63)

where $l_r=\sqrt{\hbar /m \omega _r}$ is the harmonic oscillator length in the radial plane and $\sigma _r$ and $\sigma _z$ are the dimensionless variational length parameters.

• (a) Normalize the ansatz to N atoms to show that $A=(N/2 \pi l_r^3 \sigma _r^2 \sigma _z)$.

• (b) Show that the variational energy of this ansatz is,

  $$\begin{aligned} E = \hbar \omega _r N \left( \frac{1}{6 \sigma _z^2} + \frac{1}{2 \sigma _r^2} + \frac{\sigma _r^2}{2} + \frac{\gamma }{3 \sigma _r^2 \sigma _z}\right) , \end{aligned}$$

  (4.64)

  where $\gamma = N a_\mathrm{s}/l_r$.

• (c) Make a 2D plot of the variational energy per particle, $E/N\hbar \omega _r$ (scaled by the transverse harmonic energy) as a function of the two variational length parameters, and plot this for $\gamma =-0.5$. Locate the variational solution in this 2D “energy landscape”. Repeat for $\gamma =-1$; what happens to the variational solution? By varying $\gamma$ estimate the critical value at which the solutions no longer exist (and they become prone to collapse).

4.6

Consider an object of mass M moving at velocity $\mathbf{v}_i$ which creates an excitation of energy E and momentum $\mathbf{p}=\hbar \mathbf{k}$. Show that Landau’s critical velocity, $v_c=\mathrm{min}(E/p)$, is equivalent to $dE/dp=E/p$. Compare Landau’s critical velocity for the ideal gas (dispersion relation $E(p)=p^2/2M$) against the weakly-interacting Bose gas. Finally show that in liquid helium II, Landau’s critical velocity is $v_c\approx 60~\mathrm{m/s}$. Hint: assume that near the roton minimum the dispersion relation, shown in Fig. 4.1b, has the approximate form $E(p)=\varDelta _0 +(p-p_0)^2/(2 \mu _0)$ where (at very low pressure) $\varDelta _0=1.20 \times 10^{-22}~\mathrm{J}$ is the energy gap, $p_0=\hbar k_0=2.02 \times 10^{-24}~\mathrm{kg~m/s}$ is the momentum at the roton minimum, $\mu _0=0.161~m_4$ is the effective roton mass, and $m_4=6.65 \times 10^{-27}~\mathrm{kg}$ is the mass of one $^4$He atom.