State-labeling Wannier-Stark atomic interferometers

Using cold ${}^{87}$Rb atoms trapped in a one-dimensional (1D)-optical lattice, atomic interferometers involving coherent superpositions between different Wannier-Stark atomic states are realized. Two different kinds of trapped interferometer schemes are presented: a Ramsey-type interferometer sensitive both to clock frequency and external forces, and a symmetric accordion-type interferometer, sensitive to external forces only. We evaluate the limits in terms of sensitivity and accuracy of those schemes and discuss their application as force sensors. As a first step, we apply these interferometers to the measurement of the Bloch frequency and the demonstration of a compact gravimeter.

Figure 1

Wannier-Stark ladder where ${\nu }_B$ is the Bloch frequency, ${\nu }_{\mathrm{HFS}}$ the hyperfine transition between the states $|g\rangle =|5{}^2{S}_{1/2},F=1,m_F=0\rangle$ and $|e\rangle =|5{}^2{S}_{1/2},F=2,m_F=0\rangle$ of ${}^{87}$Rb, $m$ the quantum number corresponding to the lattice sites, and ${\Omega }_{\Delta m}$ the coupling between the wells $m$ and $m\pm \Delta m$.

Figure 2

Experimental setup. Laser beams for optical trapping and Raman transitions are superimposed using dichroic mirrors. To allow for counterpropagating transitions, one of the Raman beams is retro-reflected.

Figure 3

Ramsey-Raman fringes for a lattice depth $U_0=3.9E_r$ showing evidence of transitions between up to $six$ neighboring lattice sites. Rabi envelopes are separated by the Bloch frequency ${\nu }_B=568.5$ Hz and contain interference fringes separated by $\Delta \nu =1/{\tilde{T}}_R$. Here ${\tau }_R=5$ ms and $T_R=100$ ms. The intensities of the Raman lasers and the lattice depth were set to optimize the contrast of the $\Delta m=\pm 3$ fringes. (Insert) Zoom on the $\Delta m=-3$ transition.

Figure 4

Ramsey-Raman frequency fluctuations measured on the $\Delta m=-7$ (black line) and $\Delta m=+7$ (green line) transitions for experimental parameters ${\tau }_R=120$ ms and $T_R=850$ ms. The half-difference (red thick line) cancels the hyperfine transition clock frequency drift.

Figure 5

Frequency fluctuations of ${\nu }_{\mathrm{HFS}}$ measured with a microwave Ramsey interferometer for experimental parameters ${\tau }_{\mathrm{MW}}=0.5$ ms and $T_R=1476$ ms. Fluctuations of the half-difference of the Ramsey-Raman interferometer on the $\Delta m=\pm 7$ transitions are displayed as a dashed red line for comparison; an artificial offset is present for clarity on the graph.

Figure 6

Differential light shift of the different Raman beams which is characterized by its mean light shift ${\nu }_{\mathrm{RDLS}}$ and its frequency inhomogeneities ${\sigma }_{\mathrm{RDLS}}$, measured by microwave spectroscopy with or without Raman beam out of resonance. Measurements are made for Raman pulses inducing a Rabi frequency of 27 Hz on the $\Delta m=-3$ transition. From a Gaussian fit on the microwave spectroscopy with the two Raman beams, frequency inhomogeneities are determined to be ${\sigma }_{\mathrm{RDLS}}=20$ Hz.

Figure 7

Allan standard deviation of the frequency fluctuations of ${\nu }_B$ on the $\Delta m=+7$, $\Delta m=-7$ transitions measured with a Ramsey-Raman interferometer and the computed half-difference of those two measurements canceling the hyperfine structure component, for experimental parameters ${\tau }_R=120$ ms and $T_R=850$ ms.

Figure 8

Short-term detection noise as a function of the number of measured atoms. ${\sigma }_{P_e}$ corresponds to the 1-shot Allan standard deviation of $P_e$. Each circle corresponds to a different set of experimental parameters. The quantum projection noise is illustrated with the dashed line.

Figure 9

Allan standard deviation of the frequency fluctuations of ${\nu }_{\mathrm{HFS}}$ measured with a microwave Ramsey interferometer of ${\tau }_{\mathrm{MW}}=0.5$ ms and $T_R=1476$ ms.

Figure 10

Systematic effects on the absolute frequency measurement of ${\nu }_B$ due to the intensity gradient of the dipole trap as a function of the distance to the dipole trap waist. Solid and dashed lines are linear fits to the data.

Figure 11

Basic scheme of the accordion interferometer.

Figure 12

Fringe pattern of the accordion interferometer, for Raman pulses of ${\tau }_R=140$ ms and separation time of $T=65$ ms. Left, measured; right, calculated. Resonant frequency inhomogeneities of about ${\sigma }_{\mathrm{RDLS}}=2.5$ Hz centered at ${\nu }_{\mathrm{RDLS}}=0$ Hz.

Figure 13

Typical accordion interferometer frequency fluctuations of the central fringe measured with the Raman frequency difference ${\nu }_R$ set on the $\Delta m=-3$ (black line) and $\Delta m=+3$ (green line) transitions, for a Raman $\pi$ pulse of ${\tau }_R=80$ ms and a time separation $T=205$ ms. Half-difference (red thick line) between the two signals is presented.

Figure 14

Best Allan standard deviation obtained from the half-difference frequency fluctuations of an accordion interferometer on the intersite transition $3\times {\nu }_B$ for a Raman $\pi$ pulse of ${\tau }_R=140$ ms and a time separation $T=175$ ms.

Figure 15

Distortion of the central fringe of the $\Delta m=+3$ intersite transition. The different interferometer fringe patterns are numerically calculated for different value of the mean RDLS ${\nu }_{\mathrm{RDLS}}$, and for a ${\tau }_R=140$ ms, $T=175$ ms, ${\sigma }_{\mathrm{RDLS}}=2.5$ Hz. The central fringe of the interferometer pattern is situated at $+1707$ Hz.