Parametric Amplification of Scattered Atom Pairs

We have observed parametric generation and amplification of ultracold atom pairs. A ${}^{87}\mathrm{Rb}$ Bose-Einstein condensate was loaded into a one-dimensional optical lattice with quasimomentum $k_0$ and spontaneously scattered into two final states with quasimomenta $k_1$ and $k_2$. Furthermore, when a seed of atoms was first created with quasimomentum $k_1$ we observed parametric amplification of scattered atoms pairs in states $k_1$ and $k_2$ when the phase-matching condition was fulfilled. This process is analogous to optical parametric generation and amplification of photons and could be used to efficiently create entangled pairs of atoms. Furthermore, these results explain the dynamic instability of condensates in moving lattices observed in recent experiments.

Figure 1

Dispersion curve for the optical lattice and experimental setup. (a) Band structure for a lattice depth of $V=0.5E_{\mathrm{rec}}$. The dashed line shows the free particle dispersion curve. The dispersion relation of the lattice allows two atoms with momentum $k_0$ to elastically scatter into the final momentum states $k_1$ and $k_2$. Energy and quasimomentum are conserved when $k_0$ is the average of $k_1$ and $k_2$ and the three points on the dispersion curve lie on a straight line. If $k_0$ is varied, the allowed values for $k_1$ and $k_2$ change. For values of $k_0$ below $\approx 0.55k_L$, where $k_L$ is the wave vector of the optical lattice, atoms cannot scatter elastically into different momentum states. The circles (squares) show allowed states $k_0,k_1,k_2$ for $k_0=0.66k_L$ ($0.70k_L$). As $k_0$ is increased, the final momentum states move closer together. Since the scattering occurs within the lowest band of the lattice, the final momentum is $k_2=(2k_0-k_1)\mathrm{Mod}(2k_L)$. (b) A ${}^{87}\mathrm{Rb}$ Bose-Einstein condensate is illuminated by two counter-propagating laser beams with detuning $\delta \nu$, which create a moving optical lattice. The condensate is initially held at rest. In the rest frame of the lattice, the condensate has quasimomentum $k_0=\frac{m\lambda }{2\hslash }\delta \nu$. (c) As $k_0$ was varied, we observed elastic scattering into states $k_1$ and $k_2$. (d) Absorption images for different lattice detunings, $\delta \nu$, showing parametric generation. After ramping up the lattice, the atoms were held for 10 ms at a constant lattice depth. They were then released from the trap and imaged after 43 ms of ballistic expansion. The field of view is $0.5\mathrm{mm}\times 0.3\mathrm{mm}$.

Figure 2

Parametric amplification of scattered atom pairs in a 1D optical lattice. (a) First, a 2 ms Bragg pulse was applied to the condensate. (b) The Bragg pulse seeded atoms along the long axis of the condensate with momentum $k_{\mathrm{Bragg}}=(k_a-k_b)$ in the lab frame. (c) The optical lattice was then adiabatically ramped on and applied for 10 ms. When the phase-matching condition was fulfilled, parametric amplification of atoms in the seeded state $k_1$ and its conjugate momentum state $k_2$ was observed. (d) Resonance curve showing amplification of $k_2$, when $k_1$ was seeded. Amplification occurred only when the phase-matching condition was met. For a fixed $k_{\mathrm{Bragg}}$, the resonance condition was found by varying the detuning $\delta \nu$ of the lattice. The data was taken for $k_{\mathrm{Bragg}}=0.43k_L$. The fraction of amplified atoms was obtained by subtracting images with and without the seed pulse. (e) Absorption images showing amplification of $k_1$ and $k_2$ when the phase-matching condition is met. The center of the resonance was at $\delta \nu \approx 5450\mathrm{Hz}$, close to the calculated value of $\delta \nu \approx 5350\mathrm{Hz}$. The width of the resonance is determined by the Fourier width of the Bragg pulse. Most of the scattered atoms in the third image were independent of the seed pulse.

Figure 3

Growth curve for atomic population in quasimomentum states $k_1$ and $k_2$ when the process was seeded. (a) Amplification of atoms with quasimomentum $k_1$ (solid points), and with the conjugate momentum $k_2$ (open points), when state $k_1$ was seeded. (b) Amplification of atoms in $k_1$ (solid points), and $k_2$ (open points), when $k_2$ was seeded. The values for $k_0,k_1,k_2$ were $0.66k_L$, $0.23k_L$, and $-0.92k_L$ respectively. The solid lines shows the expected gain using Eq. (1) with variable scale factors for each curve as the only free parameters.