Contrast decay in a trapped-atom interferometer

A Ramsey-Raman interferometer is used to realize a compact force sensor using cold ${}^{87}\mathrm{Rb}$ atoms trapped in a vertical optical lattice. The sensitivity and accuracy of the force measurements are discussed and the limits in short-term sensitivity evaluated. We reach a relative sensitivity on the Bloch frequency and thus on the gravity acceleration of $3.9\times {10}^{-6}$ at 1 s. We perform an experimental study of the influence of the transverse confinement onto the decay of the interferometer contrast and compare the measurements with a simple semiclassical model. It is shown that vertical gradients of the trapping potential can contribute significantly to the loss of contrast.

Figure 1

Wannier-Stark ladders of states. The two internal states are separated by the hyperfine transition frequency ${\nu }_{\mathrm{HFS}}$. The increment of energy between neighboring wells corresponds to the increment of gravitational potential energy $h\times {\nu }_B$, where ${\nu }_B$ is the Bloch frequency.

Figure 2

Ramsey-Raman fringes for a lattice depth $U_l=3.9E_r$ showing evidence of transitions between up to five neighboring lattice sites. Rabi envelopes are separated by the Bloch frequency ${\nu }_B\sim 568.5$ Hz and contain interference fringes with an interfringe of $\mathrm{\Delta }\nu =1/T_R$. Here ${\tau }_{\pi /2}=10$ ms and $T_R=150$ ms. The intensities of the Raman lasers and the lattice depth were set to optimize the contrast of the $\mathrm{\Delta }m=\pm 3$ fringes. Insert: zoom on the $\mathrm{\Delta }m=-3$ transition.

Figure 3

Ramsey-Raman fringes around –7 and +7 ${\nu }_B$ for a lattice depth of $U_l$ = 1.7 $E_r$ with $T_R$ = 900 ms and ${\tau }_{\pi /2}$ = 10 ms. The central fringe of the interferometers is indicated and the red curve is a sinusoidal fit.

Figure 4

Allan standard deviation of the half difference of the frequency fluctuations of $\mathrm{\Delta }m=+7$ and –7 transition measured with a Ramsey-Raman interferometer with experimental parameters ${\tau }_{\pi /2}$ = 10 ms and $T_R$ = 900 ms. The half difference cancels the hyperfine transition fluctuations and decays in $1/\sqrt{T_{\mathrm{meas}}}$. The black dashed line is a guide to the eyes.

Figure 5

Measurement of the Bloch frequency ${\nu }_B$ versus the IR transverse trap laser power for different IR waist positions. Black squares: waist 2 cm below the atoms. Red circles: 1 cm above. Blue triangles on the atoms. The corresponding curves are linear fits. The measurements have been realized for interferometers at $\mathrm{\Delta }m$ = $\pm$ 7 and $\pm$ 6 transitions at $U_l\sim$ 1.8 $E_R$. The pink star represents the expected value ${\nu }_{B,\mathrm{expt}}$.

Figure 6

Ramsey-Raman interferometer contrast versus $T_R$ in semilog scale. Red circles correspond to $\mathrm{\Delta }m=+6$ and –6 transitions at $U_l$ = 1.8 $E_R$, green triangles correspond to $\mathrm{\Delta }m=+3$ transition at $U_l$ = 1.6 $E_R$, and blue diamonds correspond to $\mathrm{\Delta }m=+3$ transition at $U_l$ = 3.9 $E_R$. Black squares correspond to the contrast of a Ramsey-MW interferometer. All measurements were performed at $P_{\mathrm{IR}}$ =0.5 W, and at $z=-2$ cm. Black dashed curves correspond to exponential decay fits to the data.

Figure 7

Ramsey-Raman interferometer contrast decay measurements and calculations versus $P_{\mathrm{IR}}$ for different IR waist-atoms distances. Measurements have been performed for $\mathrm{\Delta }m=+6$ and –6 transitions at $U_l\sim$ 1.8 $E_R$. Black squares correspond to a IR waist-atoms distance of –2 cm, red circles +1 cm, and for the blue triangles the waist is approximately on the atoms. Green stars correspond to a spin-echo interferometer with the IR waist on the atoms. The green dashed line represents the plateau at 0.4/s. The black, red, and blue curves correspond to decay rate calculations using the model explained in Sec. 5 for a waist-atoms distance of, respectively, $z=-2,+1$, and +0.2 cm and a waist of 145 $\mu \mathrm{m}$.

Figure 8

Interferometer contrast decay versus $U_l$. Red circles: Ramsey-Raman interferometer, blue triangles: spin-echo interferometer. Measurements were performed for $\mathrm{\Delta }m=+6$ transition, for $P_{\mathrm{IR}}$ = 1 W, the IR waist on the atoms. Red and blue curves are linear fits to the data.