Accelerating and abruptly autofocusing matter waves

We predict that classes of coherent matter waves can self-accelerate without the presence of an external potential. Such Bose-Einstein condensates can follow arbitrary power-law trajectories and can also take the form of diffraction-free Airy waves. We also show that suitably engineered radially symmetric matter waves can abruptly autofocus in space and time. We suggest different schemes for the preparation of the condensate using laser beams to imprint an amplitude or a phase pattern onto the matter wave. Direct and Fourier space generation of such waves is discussed using continuous and binary masks as well as magnetic mirrors and lenses. We study the effect of interactions and find that independently of the type and strength of the nonlinearity, the dynamics are associated with the generation of accelerating matter waves. In the case of strong attractive interactions, the acceleration is increased while the radiation reorganizes itself in the form of soliton(s).

Figure 1

Different configurations for the generation of accelerating matter waves. (a) A phase or amplitude profile is imprinted to the BEC by exposing it to a pulsed off-resonance or on-resonance laser beam, respectively, that passes through an amplitude spatial light modulator. (b) An aperture is intervened between the laser pulse and the matter wave. (c) A part of the condensate is reflected by a curved magnetic mirror.

Figure 2

Accelerating matter-wave generation via two subsequent continuous phase imprints. (a), (b) A parabolic-trajectory Airy matter wave is generated with (a) $\gamma =1$, $t_f=1$, and $\alpha =0.03$ and (b) $\gamma =3$, $t_f=2$, and $\alpha =0.03$ in Eq. (17). An accelerating BEC is shown that follows (c) a cubic trajectory with $\alpha =0.01$, $\beta =5/2$, $\gamma =1$, $t_f=1$ and (d) a power-law trajectory with exponent $q=2.5$ and $\alpha =0.01$, $\beta =8/3$, $\gamma =2$, $t_f=2$. The accelerating caustics are depicted by the curved lines, while the horizontal lines correspond to $t=t_f$ and $t=2t_f$. Note that for simplicity, the truncation function in all cases presented here is taken to be same as in the exponentially apodized Airy waves [Eq. (17)], i.e., it has the form $\exp (-\alpha x^2/{\gamma }^3)$.

Figure 3

Accelerating matter waves generated in the Fourier space by the use of binary phase imprints. In the top and bottom of the figure, the trajectories are quadratic and cubic, respectively. On the left, the matter waves are generated via a binary phase and a subsequent continuous parabolic phase, whereas on the right, the focusing phase is emulated by an equivalent of a Fresnel zone plate. (a), (b) $\gamma =2$, $t_f=3$, $\alpha =0.1$, $\beta =3$, $\kappa =5$; (c), (d) $\gamma =2$, $t_f=3$, $\alpha =0.02$, $\beta =5/2$, $\kappa =5$. The accelerating caustics are depicted by the curved lines, while the horizontal lines correspond to $t=t_f$ and $t=2t_f$. The amplitude profile of the initial condition has the form $\exp (-\alpha x^2/{\gamma }^3)$.

Figure 4

Direct matter-wave generation. In the top, middle, and bottom panels, the initial conditions are provided by, Eqs. (28), (29), (30), and (31), respectively. In the left and right panels, the resulting trajectories are parabolic and cubic and are given by Eq. (22) with $b=3/2$, $c=1/2$ and $b=5/3$, $c=1/4$, respectively. The amplitude function in all cases is $A\left(x\right)=\exp [-{(x/60)}^2]$.

Figure 5

Abruptly autofocusing matter waves via (a) direct generation using Eqs. (32) and (33) with $c=1$, $b=3/2$, $r_0=10$ and (b) using Eq. (34) with the same parameters as in (a); (c) Fourier space generation using the imprinted phase of Eq. (35) with $\kappa =2$, $\gamma =3.47$, and $\beta =3$. In all of the examples presented here, the initial amplitude profile is $\exp [-{(r/40)}^2]$. For illustration purposes, and since the density contrasts attained are large, we use a logarithmic scaling $\log (1+|\psi {|}^2)$ for the wave. In all of the abruptly autofocusing examples presented here, the density contrasts attained are of the order of ${10}^4$.

Figure 6

Typical examples of nonlinear propagation of accelerating matter waves. The initial condition is given by Eq. (36) where, in the left column, $b=3/2$, generating a parabolic trajectory, while on the right column, $b=5/3$, resulting in a cubic trajectory. In the top row [(a), (b)], the nonlinearity is switched off; in the second and third rows, the interactions are attractive; while in the fourth and fifth rows, the interactions are repulsive. In the second and fourth rows, we use a cubic nonlinearity [Eq. (5)] with $A=1$, while in the third and fifth rows, a quadratic nonlinearity is considered with $A=3$.