Figure 1

(a) An example of two low-order trajectories that contribute to the two-pulse AI (SW$j$ $=$ $j\mathrm{th}$ sw pulse, RO $=$ read-out pulse). Three momentum states are shown, ($|p_0\rangle$, $|p_0+\hslash q\rangle$, and $|p_0+2\hslash q\rangle$), corresponding to the solid, dashed, and dotted lines, respectively. The sum of the Doppler and recoil phases are indicated for each of the two states that interfere at $t=2T$. The Doppler phase difference, $q{v}_0(t-2T)$, is zero at the echo time for all initial atomic velocities, $v_0=p_0/M$. The remaining phase difference, ${\omega }_q(3t-4T)$, is a result of atomic recoil, and is equal to $2{\omega }_{q}T$ at the echo time. (b) Low-order trajectories for the three-pulse AI. Here, a third sw pulse, SW3, is applied to modify the phase of the interference at $2T$. The Doppler phase difference is zero at $t=2T$ and independent of $\delta T$ for only those trajectories that differ by $\hslash q$ after SW3 and SW2. (c) Echo energy as a function of $\delta T/{\tau }_q$ for the modified AI, where ${\tau }_q=\pi /{\omega }_q$ is the recoil period. Line shapes are shown for three different pulse areas, $u_3$, to illustrate the effect of fringe narrowing that occurs for increasing interaction strength. Here, we assume that only one ground-state magnetic sublevel contributes to the signal.Reuse & Permissions

Figure 2

(a) Optical setup for the interferometer. The glass cell has dimensions $7.6\times 7.6\times 84$ cm and is oriented along the vertical. (b) Timing diagram for the AI. The gate AOM is pulsed on to allow light for each excitation pulse produced by the $k_1$ and $k_2$ AOMs. The pulse occurring at $t=T_1+\delta T$ corresponds to the third sw pulse. The read-out pulse (which is independent of the gate AOM) and the PMT gate are turned on for $\sim 9$ $\mu$s in the vicinity of the echo time, $t=T_1+2T$. (c) Example of a two-pulse grating-echo signal (from a 10 $\mu$K ${}^{87}$Rb sample) recorded by the PMT, which corresponds to an echo energy of 130 pJ. AI pulse spacing: $T=1.06338$ ms; pulse durations: ${\tau }_1=3.8$ $\mu$s, ${\tau }_2=1.2$ $\mu$s; AI and read-out beam detunings: ${\Delta }_{\mathrm{AI}}=220$ MHz, ${\Delta }_{\mathrm{RO}}=40$ MHz; AI and read-out beam intensity: $I\sim 40$ mW/cm${}^2$.Reuse & Permissions

Figure 3

(a) Demonstration of an individual recoil measurement in ${}^{85}$Rb using the modified AI at $T=36.6656$ ms. Data are recorded in two temporal windows separated by $1128{\tau }_q\sim 36.5$ ms. The relative statistical uncertainty in ${\omega }_q$ is $\sim$180 ppb, as determined from a least-squares fit. Inset: expanded view of the fringe near $\delta T=64$ $\mu$s. (b) Eighty-two independent measurements of ${\omega }_q$ in ${}^{87}$Rb displayed in chronological order. Each data point was recorded in $\sim$10 minutes of data acquisition time, with a typical statistical uncertainty of $\sim$380 ppb. Measurements are scaled by the expected value of the recoil frequency, ${\omega }_q^{\left(0\right)}=94.77384783\left(12\right)$ rad/ms, which is based on the value of $h/M$(${}^{87}$Rb) from Ref. [11] and the $F=2\to F^{\prime }=3$ transition frequency in ${}^{87}$Rb from Ref. [34]. The dashed grid lines indicate the weighted standard deviation of 339 ppb, and the standard deviation of the mean is 37 ppb. The corresponding reduced $\chi$ squared is ${\chi }^2/\text{dof}=0.93$ for $\text{dof}=81$ degrees of freedom. The mean value, shown by the solid grid line, is $\sim$2.8 ppm below the expected value, which is due to systematic effects. AI pulse parameters: $T=45.4837$ ms, ${\tau }_1=2.2$ $\mu$s, ${\tau }_2=1.4$ $\mu$s, ${\tau }_3=3$ $\mu$s, ${\Delta }_{\mathrm{AI}}=219.8$ MHz, ${\Delta }_{\mathrm{RO}}\sim 40$ MHz, $I\sim 95$ mW/cm${}^2$.Reuse & Permissions

Figure 4

Measurements of ${\omega }_q$ as a function of $z=g{\left(2T\right)}^2/2$. The vertical axis is scaled by the expected value of ${\omega }_q^{\left(0\right)}=94.77384783$ rad/ms for excitation light at ${\Delta }_{\mathrm{AI}}=219.8$ MHz above the $F=2\to F^{\prime }=3$ transition in ${}^{87}$Rb. The variation in ${\omega }_q$ spans roughly 12 ppm, which is attributed to a combination of the spatially varying $B$-field curvature and the time-varying refractive index of the cloud. The canceling $B$ fields were set to achieve the largest signal lifetime, which in this case is $2T\sim 120$ ms. Measurements of ${\omega }_q$ at each $T$ were taken at random. Repeated measurements at the same $T$ are mostly consistent, but show a slight variation outside the statistical uncertainty indicated by the error bars. This is attributed to instability in the magnetic-field environment of the laboratory. The red curve is a fit to a third-order polynomial. AI pulse parameters: ${\tau }_1=2$ $\mu$s, ${\tau }_2=1.4$ $\mu$s, ${\tau }_3=2$ $\mu$s, ${\Delta }_{\mathrm{AI}}\sim 220$ MHz, ${\Delta }_{\mathrm{RO}}\sim 40$ MHz, $I\sim 45$ mW/cm${}^2$.Reuse & Permissions

Figure 5

Relative correction to the recoil frequency due to the refractive index as a function of the detuning of the excitation field ${\Delta }_{\mathrm{AI}}$. These curves are based on Eq. (4) with a density of $\rho ={10}^{10}$ atoms/cm${}^3$. Predictions for both ${}^{85}$Rb and ${}^{87}$Rb are shown. The detuning is plotted with respect to the $F=3\to F^{\prime }=4$ transition in ${}^{85}$Rb, and the $F=2\to F^{\prime }=3$ transition in ${}^{87}$Rb. The dashed grid lines label the location of excited states [34, 36]. The “magic” frequencies, where the relative correction crosses zero, are indicated with arrows at ${\Delta }_{\mathrm{AI}}\approx -66.4$ MHz for ${}^{85}$Rb and at ${\Delta }_{\mathrm{AI}}\approx -162.6$ MHz for ${}^{87}$Rb.Reuse & Permissions