Two-Particle Four-Mode Interferometer for Atoms

We present a free-space interferometer to observe two-particle interference of a pair of atoms with entangled momenta. The source of atom pairs is a Bose-Einstein condensate subject to a dynamical instability, and the interferometer is realized using Bragg diffraction on optical lattices, in the spirit of our recent Hong-Ou-Mandel experiment. We report on an observation ruling out the possibility of a purely mixed state at the input of the interferometer. We explain how our current setup can be extended to enable a test of a Bell inequality on momentum observables.

Figure 1

Diagram of a two-particle, four-mode interferometer. An atom pair in the entangled momentum state (1) is emitted at time $t=0$. Using Bragg diffraction on optical lattices, the four input modes are then deflected at time $t_1$, and mixed two by two at time $t_2=2t_1$ on two independent splitters $A$ and $B$, with phases ${\varphi }_A$ and ${\varphi }_B$. The interference is read out by detecting the atoms in the output modes $A_{\pm }$, $B_{\pm }$, and measuring the probabilities of joint detection $\mathcal{P}(A_{\pm },B_{\pm })$. The Bragg deflector and splitters differ from their optical analogs, because rather than reversing the incident momentum, they translate the momentum by a reciprocal lattice vector $\pm \hslash k_{\ell }$. The dashed lines show the Hong-Ou-Mandel configuration.

Figure 2

Initial velocity distribution of the emitted atom pairs in the $y\text{−}z$ plane. The color scale represents the total number of atoms detected over 1169 repetitions of the experiment inside an integration volume of $9.2\times 2.4\times 0.9(\mathrm{mm}/\mathrm{s}{)}^3$ [25]. The velocities are defined with respect to the center-of-mass velocity of the atom pairs, which was measured to be 0, 0, and $94\mathrm{mm}/\mathrm{s}$ along the $x$, $y$, and $z$ directions, respectively.

Figure 3

Normalized cross-correlation $g^{(2)}(v_z^{+},v_z^{-})$. The velocities are measured along the $z$ axis and relative to the center-of-mass velocity of the atom pairs. A sliding average was performed to reduce the statistical noise. The correlation peak is elongated along the antidiagonal because the source can emit in several pairs of modes. The width of the correlation peak along the diagonal corresponds to the diffraction limit imposed by the spatial extent of the source. The white squares show the size and position in the plane $(v_z^{+},v_z^{-})$ of the integration volumes used to obtain the points in Fig. 4 for the set of modes 1.

Figure 4

Joint detection probabilities measured at the output of the four-mode interferometer for three independent sets of momentum modes $(p,-p^{\prime })$ and $(-p,p^{\prime })$. The lower graph displays the correlation coefficient $E$. The gray line represents the zero level of this coefficient, calibrated using different combinations of the same modes for which no two-particle interference can occur by construction; the width of the line is the uncertainty on the zero level. The velocities $v_z^{+}$ corresponding to the modes $p$ are 27, 29, and $31\mathrm{mm}/\mathrm{s}$ for sets 1, 2, and 3, respectively. The velocities corresponding to $p^{\prime }$ can be deduced from Eq. (12). Averages were taken over 2218 repetitions of the experiment. Error bars denote the statistical uncertainty and are obtained by bootstrapping.