Gauss Sum Factorization with Cold Atoms

We report the first implementation of a Gauss sum factorization algorithm by an internal state Ramsey interferometer using cold atoms. A sequence of appropriately designed light pulses interacts with an ensemble of cold rubidium atoms. The final population in the involved atomic levels determines a Gauss sum. With this technique we factor the number $N=263193$.

Figure 1

Implementation of the Gauss sum factorization algorithm using cold rubidium atoms. They are launched by a double MOT system and prepared in the region $P$ by an appropriate pulse sequence in the atomic ground state. In the factorization zone $F$ the atoms interact with a sequence of pulses driving a hyperfine transition. We start with a $\pi /2$ pulse followed by a sequence of $m+1$ $\pi$ pulses with the phase ${\varphi }_k(l)$ and conclude by another $\pi /2$ pulse. The phase during the $\pi /2$ pulses is ${\varphi }_i={\varphi }_f=-90°$. A fluorescence detection in the region $D$ measures the populations in both states which determines the interference signal $c_m(l)$. We record $c_m$ for $0\le m\le M$ and the sum over $c_m$ yields the Gauss sum ${\mathcal{C}}_N^{(M)}(l)$.

Figure 2

Measured interference signals $c_m(l)$ as a function of the summation index $m$ of the Gauss sum Eq. (1) determined by the number $m+1$ of pulses in the factorization sequence for $N=263193=3\times 7\times 83\times 151$. Whereas $c_m(l)$ is approximately constant for the factor $l=151$ leading to the value ${\mathcal{C}}_N^{(14)}(151)=0.69$, the corresponding signal for the nonfactor $l=150$ oscillates, creating the small value ${\mathcal{C}}_N^{(14)}(150)=-0.02$. Here we have used the maximum number $M+1=15$ of pulses.

Figure 3

Interference signals $c_m(l)$ as a function of $m$ with and without a parabolic adaption of the pulse length shown for the factor $l=7$.

Figure 4

Experimental Gauss sum factorization based on cold atoms and exemplified by the number $N=263193=3\times 7\times 83\times 151$ using the maximum number $M+1=15$ of pulses. Here we display the absolute value of the sum ${\mathcal{C}}_N^{(14)}(l)$ over the interference signals $c_m(l)$, that is, the measured Gauss sum as a function of the trial factors $1\le l\le 200$. Dominant peaks correspond to factors of $N$. The background is produced by the nonfactors.

Figure 5

Measured contrast $\mathcal{V}$ of the factorization pattern for $N=263193$ as a function of $M$. Here $M+1$ is the maximum number of factorization pulses. Already six pulses ($M=5$) yield a contrast larger than 60%.