Composite pulses for interferometry in a thermal cold atom cloud

Atom interferometric sensors and quantum information processors must maintain coherence while the evolving quantum wave function is split, transformed, and recombined, but suffer from experimental inhomogeneities and uncertainties in the speeds and paths of these operations. Several error-correction techniques have been proposed to isolate the variable of interest. Here we apply composite pulse methods to velocity-sensitive Raman state manipulation in a freely expanding thermal atom cloud. We compare several established pulse sequences, and follow the state evolution within them. The agreement between measurements and simple predictions shows the underlying coherence of the atom ensemble, and the inversion infidelity in a $\sim 80\mu \mathrm{K}$ atom cloud is halved. Composite pulse techniques, especially if tailored for atom interferometric applications, should allow greater interferometer areas, larger atomic samples, and longer interaction times, and hence improve the sensitivity of quantum technologies from inertial sensing and clocks to quantum information processors and tests of fundamental physics.

Figure 1

Energy levels (a) for stimulated Raman transitions in ${}^{85}\mathrm{Rb}$, and (b) individual dipole-allowed Raman routes for atoms initialized in the different Zeeman $m_F$ sublevels, where the counterpropagating Raman beams are opposite-circularly polarized ${\sigma }^{+}-{\sigma }^{-}$. The relative transition strengths, calculated from the Clebsch-Gordan coefficients, normalized to the 0–0 transition, are given for each route.

Figure 2

Doppler-broadened Raman line shape [54] after the molasses phase. Measured data (circles) match a simulation (dashed) for a double Gaussian with similar populations in the central peak ($4.8\mu \mathrm{K}$) and broader background ($83\mu \mathrm{K}$). $\delta$ is the Raman detuning from the light-shifted hyperfine splitting. (Inset) Deduced atom cloud velocity distribution; $v_R$ is the Raman recoil velocity. The mean temperature is $45\mu \mathrm{K}$.

Figure 3

Upper hyperfine state population $|c_2{|}^2$ as a function of Raman interaction time $t$: (a) regular Rabi flopping; (b) Rabi flopping with rotary echoes; (c) highly-sampled data for the indicated portion of (b). Circles are experimental data; lines are numerical simulations, with (green/light-gray) and without (red/dark-gray) phase noise. An example Bloch vector trajectory, starting from $|1\rangle$ and undergoing the rotary echo ${360}_0{360}_{180}$ in the presence of pulse-length and off-resonance errors, is shown (d) before (light-blue/light-gray arrow), during (dots), and at the end of (dark-blue/dark-gray arrow), ${360}_0$; and (e) before, during and after ${360}_{180}$, at which point the Bloch vector is realigned with $|1\rangle$. The red/mid-gray arrow is the field vector, around which the Bloch vector rotates.

Figure 4

Example Bloch sphere trajectories for a Bloch vector starting from $|1\rangle$, with a Raman detuning $\delta /{\Omega }_{e}ff=0.75$, during (a) a Rabi $\pi$ pulse; and (b) the three contiguous rotations constituting a waltz sequence.

Figure 5

Measured upper state populations (circles) after various ${\sigma }^{+}-{\sigma }^{+}$ inversion sequences, as functions of Raman detuning. Simulations (lines) are for a temperature, laser intensity and sublevel splitting found by fitting, within known uncertainties of measured values, to the basic $\pi$ pulse (a) data.

Figure 6

Measured upper state populations (circles) after various ${\pi }^{+}-{\pi }^{-}$ inversion sequences, as functions of Raman detuning. Simulations (lines) are for a temperature, laser intensity and sublevel splitting found by fitting, within known uncertainties of measured values, to the basic $\pi$ pulse (a) data.

Figure 7

Measured upper state populations (circles) after various ${\sigma }^{+}\text{−}{\sigma }^{+}$ inversion sequences, as functions of interpulse interaction time. Simulations (lines) are again for parameters that reproduce the results for a basic $\pi$ (${180}_0$) pulse.

Figure 8

Predicted fidelity achievable using waltz (blue/dark-gray) and knill (green/light-gray) composite pulses, compared with a simple $\pi$-pulse (red/mid-gray), for varying velocity distribution widths ${\sigma }_v$ in units of the two-photon recoil velocity $v_R$. Dashed lines show predicted behaviour for spin-polarised atoms populating only the $m_F=0$ state, while solid lines are for an even distribution across all states $m_F=-2...+2$.

Figure 9

Schematic of the experimental setup of the Raman beams: ECDL, external-cavity diode laser; PBSC, polarizing beam-splitter cube; TA, tapered amplifier; OSA, optical spectrum analyzer; BSh, beam shaper and focusing lens. The annotation bubbles show sketches of the beam spectrum at each preparation stage. For clarity, the injection-locked preamplifier following the EOM is omitted.