Parametric Amplification of Matter Waves in Periodically Translated Optical Lattices

We observe the sudden growth of small classes of Bloch waves from atomic Bose-Einstein condensates held in periodically translated optical lattices. The effect is explained by narrowband parametric amplification of Bloch waves from noise, due to phase-matched scattering of atom pairs out of the condensate. Amplification occurs above a well-defined modulation threshold, described by dynamic shaping of single-particle band structure.

Figure 1

Absorption images of atomic interference following lattice modulation and time-of-flight expansion. (a) Atoms released at the turning point of lattice translation after a number of cycles $N_{\mathrm{cyc}}$. For short modulation times peaks occur only at $p=0,\pm \hslash k_0$. Subsequently, peaks develop at $p=\pm \hslash k_0/2$. (b) Release occurs in the 76th cycle, spaced by $1/8$th of a cycle; center of mass oscillates in phase with applied drive. (c) Simultaneous doubling in 2D lattice applying the same modulation as in (a),(b) to two orthogonal standing waves. (d) Modulation for $\delta >0$ in 1D lattice; for small amplitudes, the $p=0$ peak develops a shoulder. At larger amplitudes, (e) broad peaks are apparent between $p=0$ and $\pm \hslash k_0$. For visibility, (c) shows absorption; (a),(b),(d),(e) show atomic column densities. For (a),(b),(c), ${\omega }_v=2\pi \times 10.4\mathrm{kHz}$, $x_0=.040a$, ${\omega }_D=2\pi \times 7.6\mathrm{kHz}$, (d) ${\omega }_v=2\pi \times 11.0\mathrm{kHz}$, $x_0=.035a$, ${\omega }_D=2\pi \times 12.5\mathrm{kHz}$. (e) ${\omega }_v=2\pi \times 11.0\mathrm{kHz}$, $x_0=.037a$, ${\omega }_D=2\pi \times 12.5\mathrm{kHz}$.

Figure 2

Measured critical drive strengths and detunings (boxes) and zero-parameter comparison to those predicted by band structure and numeric evolution of the GPE for $\delta <0$, ${\omega }_v=2\pi \times 11.0\pm 0.1\mathrm{kHz}$. Solid line shows analytic result for equal dressed single-particle energy at lower band center and edge, which occurs for zero drive amplitude at $\delta =J_1+J_0$. Dashed (dotted) line shows infinite effective mass at band center (edge); between these, mass is positive both at the center and edge. Circles show onset in numeric evolution of the GPE for $J_0=0.024E_r$, $J_1=-0.28E_r$, $U_{00}=U_{10}=(4/3)U_{11}=2\times {10}^{-4}{E}_r$; ${\omega }_{\mathrm{ext}}=2\pi \times 45\mathrm{Hz}$ and $N=5\times {10}^4$. Inset: Relative peak atomic densities at $|p|=0,\hslash k_0/2$ vs drive amplitude $x_0$ for ${\omega }_D=2\pi \times 8.4\mathrm{kHz}$. Modulation pulse lengths were 10 ms.

Figure 3

Renormalized band structure for $\delta <0$ (a) and $\delta >0$ (b), obtained by diagonalizing the single-particle Hamiltonian in the RWA for the two lowest bands. Solid (dashed) lines indicate energy ${\Lambda }_q^{-}({\Lambda }_q^{+})$ of the renormalized ground (excited) band, for coupling strengths $\hslash \Omega /E_r$ of (i) 0, (ii) 0.3, and (iii) 0.9. (c) Lowest band single-particle ${\Lambda }_q^{-}$ and elementary excitation ${\overline{ϵ}}_q$ energy for three drive strengths. Solid (dashed) line shows real (imaginary) part of ${\overline{ϵ}}_q$ for $\gamma U_{00}=0.005E_r$ (red) and $0.02E_r$ (blue). Lattice depth is $V_0=10.5E_r$. (d) Growth rates in dynamically unstable regions as function of condensate and perturbation momenta, from linear stability analysis of the GPE in the lower dressed band; $\hslash \delta =-1.2E_r$ $\hslash \Omega =0.35E_r$, $\gamma U_{00}=0.02E_r$.