Role of optical channeling in contrast enhancement of echo interferometers

We investigate channeling oscillations in the contrast of a density grating formed by exposing a sample of ultracold atoms to an optical standing-wave pulse. We show that these dynamics can be imprinted on the echo signal associated with a two-pulse atom interferometer. Experimental results are supported by a simple model, which describes the grating contrast as the combination of two separate physical effects, namely, matter-wave interference and classical optical channeling resulting in Pendellösung-like oscillations. We discuss the enhancement in signal strength of echo interferometers that can be achieved by relying on these oscillations.

Figure 1

Schematic of standing-wave-induced atomic gratings in a cold sample. Momentum state diffraction of a generic ground state $|1,p\rangle$ is shown following the application of two standing-wave (SW, $\pm k$) pulses, along with the backscattering of a traveling-wave readout (RO, $k_{\mathrm{RO}}$) pulse applied after either one or two SW pulses. The lower panels show experimental signals (red points) and theoretical calculations (solid black lines) for a sample of ${}^{85}\mathrm{Rb}$ atoms with a temperature of $\sim 10\mu \mathrm{K}$ in both one- and two-pulse configurations. Each panel also shows a vertical bar denoting the small integration window used to measure the peak reflectivity.

Figure 2

(a) Sketch of pyrex trapping cell and AI layout that uses an AOM-gated-PMT for signal collection. The direction of $g$ (along $z$) is shown with an arrow. Open circles: lenses; closed circles: mirrors; CAM: CCD camera; PD: photodiode; PBS: polarizing beam splitter; $\lambda /4$: quarter wave plate. (b) Timing sequence used for AI experiments. Dashed lines indicate additional pulses used for echo experiments. Gate: Electronic gate of PMT and gating AOM; RO: traveling-wave readout pulses; AI: SW excitation pulses; RP: repump laser, TRP: trapping laser; dB/dz: magnetic field gradient.

Figure 3

Peak reflectivity of the atomic gratings in the Raman-Nath limit, described by Eqs. (1) and (2) as a function of the first pulse area, $u_1={\mathrm{\Omega }}_0^2{\tau }_1/2\mathrm{\Delta }$. The black line is a second-order Taylor expansion of the reflectivity about $u_1=0$ showing the quadratic dependence of this quantity for small pulse areas. Inset: The same quantity as a function of excitation pulse duration, plotted as a fraction of ${\tau }_{\mathrm{HO}}$ [see Eq. (3)]. Calculations are shown for a range of atom-field coupling strengths (see legend).

Figure 4

(a) Center-of-mass trajectories of atoms in the far-off resonance channeling potential [Eq. (4)] as a function of excitation length. (b) Time evolution of atomic density distribution during Pendellösung-like channeling. (c) Histogram of population distribution across two lattice periods for several channeling times. (d) Backscattered signal estimated as the $q=2$ (or $\lambda /2$) Fourier harmonic of the density distributions from (b) as a function of channeling time. The vertical bar shows the maximum predicted signal which occurs at ${\tau }_1\sim 0.3{\tau }_{\mathrm{HO}}$.

Figure 5

Peak reflectivity calculated using Eq. (5) for a variety of atom-field coupling strengths shown in the legend. (a) Results for very small excitation pulse areas demonstrating the quadratic dependence (black line). (b) Results for a wider range of pulse areas. (c) Results plotted as a function of the excitation pulse length (shown as a fraction of ${\tau }_{\mathrm{HO}}$). Simulations use $\alpha =1$ and ${\tau }_{\mathrm{RN}}=0.15{\tau }_{\mathrm{HO}}$. (d) and (e) show the same calculations as (b) and (c), including an exponential decay of the form $A{e}^{-C{\tau }_1}+B$ to incorporate the effects of experimental decoherence, with $A=1,B=0.3$, and $C={\mathrm{\Omega }}_0^2/2\mathrm{\Delta }$.

Figure 6

Peak reflectivity measured in one-pulse grating experiments using a variety of atom-field coupling strengths (${\mathrm{\Omega }}_0^2/2\mathrm{\Delta }$) shown in the legends. (a) Results for very small excitation pulse areas demonstrating the quadratic dependence (black line). (b) Results for a wider range of pulse areas. (c) Results as a function of the excitation pulse length (plotted as a fraction of ${\tau }_{\mathrm{HO}}$). The data were collected with an excitation detuning of $\mathrm{\Delta }=390$ MHz. (d) Ratio of the steady-state and maximum reflectivities as a function of atom-field coupling. The solid line is an exponential fit to the data. (e) Maximum peak reflectivity attained as a function of the atom-field coupling. Here the excitation detuning was varied between $\mathrm{\Delta }=160$ and $\mathrm{\Delta }=390$ MHz to further change the atom-field coupling. The solid line shows the prediction of Eq. (5) with $\alpha =0.7$ and ${\tau }_{\mathrm{RN}}=0.18{\tau }_{\mathrm{HO}}$.

Figure 7

Peak reflectivity measured in two-pulse grating echo experiments using a variety of atom-field coupling strengths shown in the legends. (a) Results for very small excitation pulse areas with a quadratic fit (black line). (b) Results for a wider range of pulse areas. (c) Results as a function of the excitation pulse length (plotted as a fraction of ${\tau }_{\mathrm{HO}}$). (d) Ratio of steady-state and maximum reflectivities as a function of atom-field coupling. The solid line is an exponential fit to the data. (e) Maximum peak reflectivity attained for each atom-field coupling. The solid line shows the predictions of Eq. (5) with $\alpha =1$ and ${\tau }_{\mathrm{RN}}=0.20{\tau }_{\mathrm{HO}}$ chosen to match the experimental data. Here, pulse separation $T=70.2\mu \mathrm{s}$, second pulse duration ${\tau }_2\approx 0.15{\tau }_{\mathrm{HO}}$, and excitation detuning $\mathrm{\Delta }=390$ MHz.

Figure 8

Peak reflectivity of grating echo experiments as a function of the second pulse duration for various atom-field coupling strengths (shown in the legend). Results are plotted as a function of (a) the pulse area and (b) the pulse duration as a fraction of ${\tau }_{\mathrm{HO}}$. In (b), the vertical line shows the predicted end of the Raman-Nath regime, defined by ${\tau }_2\sim 0.2{\tau }_{\mathrm{HO}}$. These data were collected using $T=70.2\mu \mathrm{s}$, a first excitation pulse of length ${\tau }_1=0.3{\tau }_{\mathrm{HO}}$, and an excitation detuning of $\mathrm{\Delta }=160$ MHz.