Tunable source of correlated atom beams

We use a one-dimensional optical lattice to modify the dispersion relation of atomic matter waves. Four-wave mixing in this situation produces atom pairs in two well-defined beams. We show that these beams present a narrow momentum correlation, that their momenta are precisely tunable, and that this pair source can be operated in the regimes of low mode occupancy and of high mode occupancy.

Figure 1

(a) 1D pair creation process in an optical lattice with period ${\lambda }_{\mathrm{latt}.}/2$: The dispersion relation in the first Bloch band (green solid curve) allows scattering of atoms from a BEC with quasimomentum $k_0$ (open red circle) in the lattice frame into pairs with quasimomenta $k_1$ (filled orange circle) and $k_2$ (filled blue circle), so that phase-matching conditions given by energy and momentum conservations are fulfilled. The example here is for a lattice depth $V_0=0.725E_{\mathrm{rec}}$ and $k_0=-0.65k_{\mathrm{rec}}$, with $k_{\mathrm{rec}}=2\pi /{\lambda }_{\mathrm{latt}.}$ the recoil momentum and $E_{\mathrm{rec}}={\hslash }^2{k}_{\mathrm{rec}}^2/2m=h\times 44$ kHz the recoil energy. (b) Vertical single-shot momentum distribution (integrated over the total transverse distribution) measured for these conditions. The three main peaks correspond to the initial BEC and to the macroscopically populated beams centered at $k_1$ and $k_2$, which are mainly projected in the first Brillouin zone (in white) when the lattice is switched off. As expected, small diffraction peaks at $k_0+2k_{\mathrm{rec}}$ and $k_2-2k_{\mathrm{rec}}$ are also visible, due to the proximity of $k_0$ and $k_2$ to the band edge.

Figure 2

Experimental setup and sequence: (1) Initially, a BEC of metastable helium is trapped in a vertical optical potential with a 43 $\mu$m waist. (2) An optical lattice is suddenly applied in the presence of the trap. It is tilted by 7${}^{\circ }$ with respect to the trap axis, and is focused on the BEC with a 200 $\mu$m waist. (3) After the dipole trap and optical lattice switch off, the cloud expands and falls on the 3D resolved single atom detector. Given the values of the vertical and transverse Thomas-Fermi radii (0.5 mm and 3 $\mu$m), the arrival time and position reflect the 3D momentum distribution, provided the momenta are well above $3\times {10}^{-2}{k}_{\mathrm{rec}}$ along $z$ and $2\times {10}^{-4}{k}_{\mathrm{rec}}$ transversely.

Figure 3

(a) and (b) Cuts along $y$ and $z$ of the integrated, normalized cross-correlation function of the two beams, $g_C^{\left(2\right)}\left(\Delta \mathbf{k}\right)=\int d{\mathbf{k}}_{\mathbf{i}}{g}_C^{\left(2\right)}({\mathbf{k}}_{\mathbf{i}},{\mathbf{k}}_{\mathbf{j}}+\Delta \mathbf{k})$. The integration over the momentum distribution ${\mathbf{k}}_{\mathbf{i}}$ is performed on a box with dimensions $L_{k_x}=L_{k_y}=0.4k_{\mathrm{rec}}$ and $L_{k_z}=5\times {10}^{-2}{k}_{\mathrm{rec}}$ centered on beam 1, ${\mathbf{k}}_{\mathbf{i}}+{\mathbf{k}}_{\mathbf{j}}=(k_1+k_2)\widehat{{\mathbf{e}}_{\mathbf{z}}}$, and the cuts have a thickness ${10}^{-2}{k}_{\mathrm{rec}}$ ($1.5\times {10}^{-1}{k}_{\mathrm{rec}}$) along $z$ ($x$ and $y$). The bunching, due to the correlation between the two beams, has a longitudinal (transverse) width ${\sigma }_{c,z}=1.8\times {10}^{-2}{k}_{\mathrm{rec}}$ (${\sigma }_{c,y}=1.6\times {10}^{-1}{k}_{\mathrm{rec}}$). (c) and (d) Cuts along $y$ and $z$ of the integrated, normalized local correlation function of beam 1, $g_L^{\left(2\right)}\left(\Delta \mathbf{k}\right)=\int d{\mathbf{k}}_{\mathbf{i}}{g}_L^{\left(2\right)}({\mathbf{k}}_{\mathbf{i}},{\mathbf{k}}_{\mathbf{i}}+\Delta \mathbf{k})$. The integration region is the same as for the cross correlation, and the cuts have a thickness $2.5\times {10}^{-3}{k}_{\mathrm{rec}}$ ($0.1k_{\mathrm{rec}}$) along $z$ ($x$ and $y$). The bunching, due to the HBT effect, has a longitudinal (transverse) width ${\sigma }_{l,z}=3.7\times {10}^{-3}{k}_{\mathrm{rec}}$ (${\sigma }_{l,y}=1.3\times {10}^{-1}{k}_{\mathrm{rec}}$). Cuts along $x$ (not shown here) have the same widths and amplitudes as cuts along $y$. These correlation functions are calculated using 850 experimental realizations, with $k_0=-0.65k_{\mathrm{rec}}$, a lattice depth $V_0=0.725E_{\mathrm{rec}}$, and a lattice duration $T_L=2$ ms. In all plots, the horizontal error bars indicate the bin size and the vertical ones correspond to the statistical $1\sigma$ uncertainties. The solid lines are Gaussian fits to the data from which we extract the correlation widths.

Figure 4

Measured mean momenta $k_1$ and $k_2$ of the beams (black dots, in units of $k_{\mathrm{rec}}$) as a function of $k_0$ (initial BEC momentum in the lattice frame) for a depth $V_0=1.05E_{\mathrm{rec}}$ and a duration $T_L=1.5$ ms of the lattice. The solid line shows the phase-matching curve expected without interactions, while the dashed line includes the mean field [see Eq. (2)].

Figure 5

Dependence of the population of beam 1 (orange filled circles) and beam 2 (blue open circles) on the lattice duration $T_L$ for $k_0=-0.67k_{\mathrm{rec}}$ and for a lattice depth $V_0=1.05E_{\mathrm{rec}}$. The gray line is an exponential fit of the detected population in beam 2 for $T_L<0.3$ ms, which gives a time constant of 0.1 ms and an offset of 11.5 detected atoms. This offset is due to the small thermal part of the source cloud with quasimomenta $k_1$ and $k_2$. For a lower lattice depth, as for the data of Fig. 3, the temporal evolution is a few times slower [21]. Inset: same data with linear scale.

Figure 6

Normalized variance of atom number difference between two regions selected close to beams 1 and 2. The data are the same as those of Fig. 3. Regions are vertical cylinders of radius $2.5\times {10}^{-2}{k}_{\mathrm{rec}}$ and height $8.5\times {10}^{-2}{k}_{\mathrm{rec}}$. They are centered on the two beams in the transverse plane. Along the vertical axis, the center momentum (in the lattice frame) of region 1 is scanned, whereas region 2 is fixed. A variance below unity indicates sub-Poissonian fluctuations.