Enhanced Factoring with a Bose-Einstein Condensate

We present a novel method to realize analog sum computation with a Bose-Einstein condensate in an optical lattice potential subject to controlled phase jumps. We use the method to implement the Gauss sum algorithm for factoring numbers. By exploiting higher order quantum momentum states, we are able to improve the algorithm’s accuracy beyond the limits of the usual classical implementation.

Figure 1

The setup for implementing analog sum computation. An acousto-optic modulator (AOM) is driven by a pulse as shown in (a) consisting of an 80 MHz sine wave subject to phase jumps ${\varphi }_i$ separated by intervals $|a_i|$. Light from a single laser source is split into two beams, one of which passes through the AOM driven by the signal with phase jumps. The two beams then impinge on the BEC in couterpropagating configuration to create a standing wave as seen in (b).

Figure 2

Output from the TGS algorithm for $N=91$. Absorption images (with background suppression) are shown in (a), demonstrating the diffraction pattern which constitutes the factoring signal. (Note that background “speckle” is also visible in some images). (b) shows the Energy measurements derived from the raw data (circles) which are compared with the 10 term TGS from Eq. (4) (dashed line). Crosses show energy predictions for the quantum system as given by Eq. (3) which agree perfectly with the TGS. Note that the scaled energy $E$ is dimensionless.

Figure 3

Experimental measurements are compared with theory for (a) $N=231$ and (b) $N=17947$. In both cases a partial factorization is presented. Circles with error bars show experimental data, while dashed red lines show the theoretical predictions. Note that the TGS only exists for integer $l$ and the line joining the points is merely to aid the eye. In both cases, we found excellent agreement with the theory of Eq. (4).

Figure 4

Demonstrations of enhanced factoring accuracy (as measured by visibility $V$) using $P_n$ when $N=231$ [(a),(c)] and $N=17947$ [(b),(d)]. (a) and (b) show $V$ as a function of $M$ for the standard TGS (solid line) and $V_n^{\prime }$ for $n=2$ (dashed line),$n=4$ (dashed-dotted line) and $n=6$ (dotted line). The lower row shows experimental partial factorizations (black circles with error bars) for the same case as Fig. 3 but using the enhanced method with $n=2$. The solid black line shows the theoretical prediction $P_n$, while the red (dashed) line shows the usual Gauss sum. Note that the two different quantities can be compared since they are both dimensionless and normalized to 1.