Figure 1

Recoil diagrams for the two-pulse (a) and three-pulse (b) AIs in the absence of any external forces. Standing wave excitations are labeled by SW1–SW3. At $t=T_1$ the atom is diffracted into a superposition of momentum states differing by integer ($n$) multiples of $\hslash q$. A second sw pulse is applied at $t=T_2$ which further diffracts the atoms. In the two-pulse case, interference between states differing by $\hslash q$ ($\Delta n=\pm 1$) occurs at times $t_{\mathrm{echo}}^{\left(2\right)}=T_1+(\overline{N}+1)T_{21}$. For the three-pulse scheme, a third sw pulse is applied at $t=T_3$ which produces interference at times $t_{\mathrm{echo}}^{\left(3\right)}=T_1+(\overline{N}+1)T_{21}+T_{32}$. Examples of interfering trajectories for $\overline{N}=1$ and 2 are labeled by solid black and dashed blue lines, respectively. Circles indicate locations where interference fringes occur with spatial frequency $q$.Reuse & Permissions

Figure 2

The predicted $B$-gradient signal for the two-pulse AI (a) and (b) [based on Eq. (7)], where $T_{21}$ is varied in steps of ${\tau }_q\sim 32$ $\mu$s, and the three-pulse AI (c) and (d), where $T_{32}$ is varied. In all plots the solid red line is the real part of the scattered field, the black dashed line shows the field intensity (which is measured in the experiment), the $B$ gradient is fixed at $\beta =10$ mG/cm, and the first-order echo ($\overline{N}=1$) is used. In parts (a) and (c) the excitation beams are circularly polarized ($|q_L|=1$) and the magnetic sublevel populations are equally distributed among the $F=3$ ground state of ${}^{85}$Rb. In this case, the field undergoes amplitude modulation with multiple frequency components, and the field intensity exhibits modulation with a contrast $<100\%$. For the two-pulse AI (a), the field intensity exhibits modulation at a chirped frequency, whereas for the three-pulse AI (c) the modulation is not chirped. Parts (b) and (d) show the field generated by a sample that is optically pumped equally into the $|3$$-3\rangle$ and $\left|33\right.〉$ states with linearly polarized excitation beams ($q_L=0$). Here, both the field and the field intensity exhibit modulation with only one frequency component, and the contrast of the oscillations is 100$\%$. For both (c) and (d), $T_{21}$ was fixed at a typical experimental value of 5 ms.Reuse & Permissions

Figure 3

Diagram of the experiment. The excitation beams are both ${\sigma }^{+}$ polarized by the $\lambda /4$ plates. The glass cell has approximate dimensions $7.6\times 7.6\times 84$ cm. Each pair of square quadrupole coils has a side length of $\sim 66$ cm and contain overlapped coils wired in both Helmholtz and anti-Helmholtz configurations for canceling $B$ fields and $B$ gradients, respectively.Reuse & Permissions

Figure 4

Schematic of the rf chain used for chirped AI pulses. A phase-locked loop (PLL) generates a 220-MHz rf signal which is split and mixed with the output of two separate arbitrary waveform generators (AWGs). The AWGs are triggered at the start of the experiment to output a frequency sweep from 20 MHz to $20\text{MHz}\pm \delta \left(t\right)$ after a time $t$, where $\delta \left(t\right)=gt/\lambda$, $g\sim 9.8$ m/s${}^2$ and $\lambda \sim 780$ nm. The sum frequency from the mixers is isolated using a band-pass filter (BPF) with a center frequency of 240 MHz and a 5-dB pass band of 4 MHz. The outputs of the BPFs are pulsed using a set of transistor-transistor logic (TTL) switches, which are then sent to the ${\mathit{k}}_1$ and ${\mathit{k}}_2$ AOMs. The two-pulse AI sequence for both ${\mathit{k}}_1$ and ${\mathit{k}}_2$ are shown. Here, P1 and P2 refer to traveling wave components comprising the first and second sw pulses, and RO denotes the traveling wave read-out pulse sent along ${\mathit{k}}_1$. Both AWGs and the PLL are externally referenced to a 10-MHz Rb clock.Reuse & Permissions

Figure 5

(a) Temperature measurement of the laser-cooled sample. The horizontal $e^{-1}$ cloud radius is measured as a function of expansion time using a CCD camera. A hyperbolic fit to these data (shown as the solid red line) yields a measurement of $\mathcal{T}=2.4\pm 0.2$ $\mu$K. Each error bar is the $1\sigma$ statistical uncertainty of the cloud radius obtained from a Gaussian fit to each cloud profile. (b) Data showing the signal lifetime for various pulse configurations. The horizontal axis is the time of the read-out pulse $T_{\mathit{RO}}$, relative to the time of trap turn-off $T_0$, which signifies the start of the experiment. An echo energy of $\sim 0.1$ pJ is equivalent to the level of noise in the detector. These data were obtained with pulse durations for the two-pulse (three-pulse) AI: ${\tau }_1=800$ (800) ns, ${\tau }_2=100$ (70) ns, ${\tau }_3=0$ (70) ns; pulse intensity $I\sim 64$ mW/cm${}^2$; and a sample temperature of $\mathcal{T}\sim 20$ $\mu$K. Pulse separations for each configuration are shown in the figure. (c) Pulse timing diagrams for the data shown in (b). The red arrows distinguish which pulses were varied and the time step $dT$ indicates the amount each pulse was incremented relative to the others. For the two-pulse AI, $dT=n{\tau }_q$, where $n=10$. Pulses without arrows are fixed in time.Reuse & Permissions

Figure 6

(a) First order ($\overline{N}=1$) two-pulse echo signal for various applied $B$-field gradients $\beta$. The theoretically expected echo time is at $\Delta t=t-2T_{21}=0$. Each mA of current corresponds to a change of $\sim 0.04$ mG/cm in the applied gradient. (b) First order ($\overline{N}=1$) two-pulse echo energy as a function of $\beta$. The solid red line is a fit based on Eq. (12). The value of $\beta$ is obtained from a calibration using a flux-gate magnetometer. Pulse parameters for both (a) and (b): ${\tau }_1=800$ ns, ${\tau }_2=130$ ns, $I\sim 64$ mW/cm${}^2$, $T_{21}\sim 40.6$ ms.Reuse & Permissions

Figure 7

(a) Two-pulse echo energy as a function of $T_{21}$ in the presence of a fixed $\beta$. Gradient-induced oscillations are shown for the first two echo orders (black points, $\overline{N}=1$; gray points, $\overline{N}=2$). Here, $T_{21}$ is varied in integer multiples of ${\tau }_q$ to avoid sensitivity to atomic recoil effects. Fits based on Eq. (12), shown as the solid lines, give $\left|\beta \right|=9.30\left(1\right)$ mG/cm and $\left|\beta \right|=9.34\left(1\right)$ mG/cm, respectively. (b) Data analogous to (a) obtained using a slightly larger gradient with the three-pulse AI. Here, $T_{21}$ is fixed at 2.0 ms and $T_{32}$ is varied to map out the modulation for the first two echo orders (black points, $\overline{N}=1$; gray points, $\overline{N}=2$). Fits based on Eq. (13) give $\left|\beta \right|=17.50\left(4\right)$ mG/cm and $\left|\beta \right|=17.78\left(5\right)$ mG/cm, respectively. Two-pulse (three-pulse) AI parameters: ${\tau }_1=800$ (800) ns; ${\tau }_2=100$ (70) ns; ${\tau }_3=0$ (70) ns; $I\sim 64$ mW/cm${}^2$.Reuse & Permissions

Figure 8

Smallest direct measurement of a $B$ gradient using the two-pulse (a) and three-pulse (b) AIs. Here, the applied fields were tuned separately for each AI such that the first revival in the $\overline{N}=1$ echo energy occurred at the largest time. A fit based on Eq. (12) for the two-pulse AI gave $\left|\beta \right|=0.26\left(3\right)$ mG/cm. Similarly, a fit based on Eq. (13) yielded $\left|\beta \right|=9.5\left(1\right)$ mG/cm for the three-pulse AI, with $T_{21}\sim 1.27$ ms. Two-pulse (three-pulse) AI parameters: ${\tau }_1=800$ (800) ns, ${\tau }_2=100$ (70) ns, ${\tau }_3=0$ (70) ns, $I\sim 64$ mW/cm${}^2$.Reuse & Permissions

Figure 9

The real part of the scattered electric field (red solid line) in the presence of gravity for the two-pulse (a) and (b) and the three-pulse (c)–(f) AIs. In parts (a)–(d), the black dashed line shows the signal envelope [based on Eq. (A21) for (a) and (b) and Eq. (A29) for (c) and (d)], which is modulated at $2{\omega }_q\approx 2\pi \times 31$ kHz. (a) The field oscillates at a frequency ${\omega }_g^{\left(2\right)}\left(T_{21}\right)\sim 2\pi \times 50$ kHz in the vicinity of $T_{21}\sim 1$ ms. (b) The modulation frequency increases to ${\omega }_g^{\left(2\right)}\sim 2\pi \times 450$ kHz at $T_{21}\sim 9$ ms. Parts (c) and (d) show results for the three-pulse AI as a function of $T_{21}$, with $T_{32}$ fixed at 100 $\mu$s. The field oscillates at approximately the same frequency in (a) and (c), and in (b) and (d), since $T_{32}$ is small. Parts (e) and (f) show the three-pulse signal as a function of $T_{32}$ with $T_{21}=5$ ms. Since $T_{21}$ is fixed, there is no sensitivity to atomic recoil and the signal envelope is not modulated. As $T_{32}$ increases, the modulation frequency of the scattered field amplitude remains fixed at ${\omega }_g^{\left(3\right)}\left(T_{21}\right)\sim 2\pi \times 125$ kHz.Reuse & Permissions