Transverse laser modes in Bose-Einstein condensates

Bose-Einstein condensates (BECs) are in many ways similar to laser light. In Fox-Li-type simulations, which involve free evolution of a two-dimensional BEC in combination with periodic focusing and localized loss, we model here numerically the BEC equivalent of transverse aspects of laser resonators. We discuss possible experimental realizations in the form of periodic series of light pulses interacting with the BEC. We show that such experiments can result in trapped, structurally stable, modes analogous to Hermite-Gaussian laser modes. This outlines a new way of trapping BECs and illustrates further the analogy between BECs and laser light.

Figure 1

Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes. Shown here are the intensity cross sections of two HG modes, ${\text{HG}}_{2,0}$ and ${\text{HG}}_{2,1}$, and the intensity and phase (gray scale representation) cross sections of one LG mode, ${\text{LG}}_{4,0}$, which possesses a central charge-4 vortex.

Figure 2

Laser resonator (a) and its unfolded lens-series equivalent (b). A light beam passing through the lens-series equivalent is periodically being focused, just like light in the resonator is being focused every time it is reflected off one of the resonator’s mirrors. A series of “lens pulses”—short, off-resonant light pulses with a specific intensity profile—can repeatedly focus a BEC in a similar way (c). For laser modes to form localized absorption needs to occur in the resonator; this is not shown here. Note the role reversal: in (a) and (b), a light pulse is being focused (by mirrors and lenses, respectively); in (c) light pulses focus the BEC.

Figure 3

Focusing of a BEC by a lens pulse. A BEC in the ground state was focused using a light pulse of focal time $f$, which phase shifts the BEC locally by $r^2∕\left(2f\right)$. The graph shows the actual focusing time $t_f$, which is the time it takes the BEC to become phase flat after interaction with the lens pulse, as a function of $f$. Analogy with ray optics suggests that the BEC comes to a focus after a time $t_f=f$; however, to our surprise we find that this is only the case for small values of $f$. In the range $0⩽g⩽100$, over which our simulations were performed, these results are virtually independent of $g$.

Figure 4

BEC after interaction with $p$ lens pulses and absorption events of a light-pulse resonator (the value of $p$ for each frame is shown in the top-left corner), starting with the harmonic trap ground state $\left(p=0\right)$, calculated for $g=100$. The pulses of focal time $f=0.5$ are separated by $L=0.2$ (in dimensionless time units); the density patterns shown here are calculated a time $L∕2$ after the actual pulse, i.e., half time between the previous and the next pulse. Note that the bright central spots for $p=2,3$, and 7 have no special significance other than that the BEC happens to be very dense in the center at the particular time during the respective resonator periods these density snapshots were taken. The curve at the bottom shows the fraction of the initial number of particles, $N_0$, that remains in the BEC after $p$ periods of the light-pulse resonator. To demonstrate the loss, the density is not normalized in the frames in this figure; the full scale of grays represents in the frames for $p=0$ to 4 densities between 0 and 0.2, in the remaining frames between 0 and 0.05.

Figure 5

Density patterns of some eigenmodes of the standard light-pulse resonator with a nonlinearity of $g=30$. By varying the cross-hair position, different patterns—characterized here by their number of nodal lines in the horizontal and vertical direction, $n$ and $m$, respectively—emerge, which strongly (but not exactly) resemble optical Hermite-Gaussian modes. These pictures show the BEC halfway between two lens pulses. A possible position of the cross hair (white dashed lines) is shown in the two rightmost frames. The pictures show only the central $6.25\times 6.25$ units of the modeled area.

Figure 6

Density (top) and corresponding phase patterns (below) for various values of the nonlinear coefficient, $g$. In all cases, the shape of the absorption region is a narrow vertical straight line.

Figure 7

Free time evolution of an eigenmode, calculated for $g=0$. If—after the eigenmode has formed—the series of light pulses is switched off at $t=0$ (the last few pulses arrive at $t=-2L,-L,0$), the BEC approximately retains its shape but changes its size. The graph shows the distance from the center of the maximum of the density, which indicates the relative size; the points do not lie on a smooth curve due to numerical errors. In analogy with optical higher-order Hermite-Gaussian modes, the BEC passes through a focus, where it is smallest. On either side of the focus, its size changes approximately symmetrically. The data points for $t<0$ were calculated using “backward propagation” in time. The graph is calculated for $L=0.2$ (in adimensional time units); the vertical range then represents three adimensional length units. To avoid numerical artifacts in obtaining this graph, the BEC was simulated over an area larger than the standard $16\times 16$ nondimensional units.

Figure 8

Density patterns for the output of different configurations of resonator, calculated for $g=0$. $f$ represents the focal time of the lens pulse, $L$ the time between subsequent pulses (both in dimensionless units). For the particular choice of loss-pulse shape, the loss per period of the light-pulse resonator for these eigenmodes is, from top left to bottom right, $1.8\%$, $1.3\%$, $5.9\%$, $4.4\%$, $10.2\%$, and $9.7\%$.

Figure 9

Modeled density (first and third picture row) and phase (second and fourth picture row; gray-level representation) immediately after a BEC initially in an eigenmode with $n$ horizontal and $m$ vertical nodal lines has interacted with a pair of cylindrical-lens pulses. The intensity patterns show strong signs of the ring structure characteristic of LG modes ($p$ bright rings and a central bright spot if $l=0$, $p+1$ bright rings if $l\ne 0$), although this ring structure is not perfect and becomes less clear for larger values of $g$ (see $g=50$, bottom row). The phase patterns show the presence of, respectively, $l$ vortices within the innermost intensity ring, situated at positions where all possible phases meet (marked by arrows). From top left to bottom right, the parameters of the initial eigenmode and lens pair($f$, focal time of each lens; $s$, separation between lenses; both were optimized in each case to give visually the best ring structure) are as follows: $n=1$, $m=0$, $f=0.46$, $s=0.35$; $n=2$, $m=2$, $f=0.25$, $s=0.18$; $n=6$, $m=4$, $f=0.25$, $s=0.18$; $n=1$, $m=0$, $f=0.23$, $s=0.35$; $n=2$, $m=0$, $f=0.23$, $s=0.3$; $n=4$, $m=0$, $f=0.1$, $s=0.11$.