Controlling matter waves in momentum space

The transformation design method of momentum for matter waves in a harmonic trap is proposed. As applications, we design (1) a momentum invisibility cloak to control the distribution of a wave function in momentum space, (2) a quantum localization cloak that localizes a matter wave around zero momentum, and (3) the unusual quantum states of momentum space. Comprehension of these momentum cloaks in position space through the Fourier transformation is presented. In contrast to the construct of quantum cloaks in position space, the momentum cloaks presented here can only be reached by controlling the spring parameter of the trap and offering a potential there, without needing to control the effective mass of quantum particles themselves. The presented discussions also provide a possible inspiration to help localize and maintain a quantum state in momentum space by way of controlling the shape of a trap and a supplied potential.

Figure 1

Cloaking of matter waves in momentum space. The pattern shows the perfect cloaking of matter waves in the $x\text{−}y$ plane of momentum space by the momentum cloak made from the specific spring parameters of a harmonic trap and a potential in Eqs. (18) and (19), where the radius of the inner (outer) shell of the cloak was chosen as $p_a=0.4\left(p_b=0.5\right)$; the wave length of the matter wave is chosen as ${\lambda }_p=p_a$.

Figure 2

Alternative viewpoint of quantum cloaking in momentum space. The pattern exhibits how the probability current in momentum space was perfectly guided to flow around the invisible region around the origin $p=0$. The blue lines are the streamlines of the probability current.

Figure 3

The parameter $k_{11}\left(r\right)$ of the momentum cloak in position space. The curve shows the Fourier transformation of the spring parameter ${\overline{k}}_{11}\left(p\right)$ of the presented momentum cloak, where the transverse and longitudinal axes are measured by the units $r_a=\hslash /p_a$ and $k_0=\overline{k}{p}_a^3$. The position space viewpoint provides us with the chance to realize the momentum cloak by the manner of position space.

Figure 4

The parameter $k_{22}\left(r\right)$ of the momentum cloak in position space. The presented curve shows the position space version of the spring parameter ${\overline{k}}_{22}\left(p\right)$ of the momentum cloak. It shows that the main influence of ${\overline{k}}_{22}\left(p\right)$ is only in the finite range $r<20r_a$.

Figure 5

The position space version of the effective potential in the cloaking shell of the momentum cloak. The curve is the Fourier transformation of $U\left(p\right)$ needed in the cloaking shell for the perfect cloaking of matter waves in momentum space, where the longitudinal axis is normalized by $U_0=E{p}_a^3$. The curve shows that the dominant influence of the potential is in the finite range $r<20r_a$.

Figure 6

The effect of the localization cloak of momentum. The pattern shows the invisibility localization effect of the localization cloak presented in Sec. 3 B for a uniform distribution of matter waves in momentum space. Here the inner (outer) shell of the cloak is chosen as $p_a=1\left(p_b=2\right)$, and the wave length of the matter waves is taken as ${\lambda }_p=p_a$ for exhibiting the localization effect clearly. The range of the uniform wave distribution has been assumed to be 10 wavelengths.

Figure 7

The parameter $k_{11}\left(r\right)$ of the localization cloak in position space. The damping-type curve is the Fourier transformation of the parameter ${\overline{k}}_{11}\left(p\right)$ for the momentum localization cloak presented in Sec. 3 B with $p_a=0.4$ and $p_b=0.5$. It shows us the version of ${\overline{k}}_{11}\left(p\right)$ in position space. Physically it exhibits a periodic pushing and pulling force on the particle.

Figure 8

The parameter $k_{22}\left(r\right)$ of the localization cloak of momentum in position space. The curve shows the spring parameter ${\overline{k}}_{22}\left(p\right)$ in position space. It is used to establish the localization cloak of momentum in position space, and is given by the Fourier transformation of ${\overline{k}}_{22}\left(p\right)$. It shows that the action of ${\overline{k}}_{22}\left(p\right)$, and thus ${\overline{k}}_{33}\left(p\right)$, decays rapidly and approaches zero after $r>20r_a$.

Figure 9

The position space version of the effective potential required for invisibly localizing a matter wave with the localization cloak of momentum. The curve shows the position space behavior of the effective potential within the localization cloak of momentum. It is given by the Fourier transformation of $U\left(p\right)$. The curve shows that the effective potential offers a strong attractive force to the particle in $r<4r_a$ to reach the localization effect.