Atom interferometry using a shaken optical lattice

We introduce shaken lattice interferometry with atoms trapped in a one-dimensional optical lattice. By phase modulating (shaking) the lattice, we control the momentum state of the atoms. Through a sequence of shaking functions, the atoms undergo an interferometer sequence of splitting, propagation, reflection, reverse propagation, and recombination. Each shaking function in the sequence is optimized with a genetic algorithm to achieve the desired momentum state transitions. As with conventional atom interferometers, the sensitivity of the shaken lattice interferometer increases with interrogation time. The shaken lattice interferometer may also be optimized to sense signals of interest while rejecting others, such as the measurement of an ac inertial signal in the presence of an unwanted dc signal.

Figure 1

Full interferometer sequence. Blue clouds represent atom wavepackets interacting with the shaken lattice. Atoms begin in the ground Bloch state of the lattice (Fig. 5) and are split into two oppositely propagating wavepackets. The atoms are then reflected, reverse-propagated, and recombined back into their initial ground state (in the absence of a signal), thus completing the interferometer sequence.

Figure 2

Reciprocal interferometer sequence. The reciprocal interferometer modifies the standard Michelson interferometer sequence shown in Fig. 1 so that the atoms travel a fully symmetric path. This configuration is designed to be sensitive to ac accelerations and immune to dc accelerations.

Figure 3

Block diagram illustrating the steps taken in the GA. Given the initial and desired states, the first generation $G=1$ of $A$ individuals is randomly generated with unique shaking functions ${\varphi }_{\alpha }\left(t\right)$. We then solve the TDSE for each individual in the generation. These results are then used to produce the next generation of individuals, denoted $G=2$. After the $j\mathrm{th}$ run of the simulation a generation $G=j+1$ results from the mixing of the previous generation's individuals (see Fig. 4). Once the convergence criterion is met, the GA stops.

Figure 4

Each iteration, the genetic algorithm mixes the best individuals from the previous generation to create “children” that populate the next generation. Each individual is a vector of values where red corresponds to a minimum value and purple corresponds to a maximum value. In each case the indices of crossover or mutation are chosen at random. The numbers label the index and their color is chosen for legibility. The methods presented here are adapted from Ref. [1]. (a) One-point crossover at index 6. (b) Two-point crossover at indices 5 and 9. (c) Random mutation of the value at index 2. (d) Change of the value at index 5 through a small “creep” of the value.

Figure 5

Quantized momentum population of atoms in the ground Bloch state of an infinite lattice with $V_0=10E_{\mathrm{R}}$. This ground state is the initial wavefunction of the atoms at the start of the interferometer sequence and the final state of the atoms after recombination with no signal applied.

Figure 6

Shaking protocol for the optimal splitting shown in Table 1. The envelope function ensures slow turn on and turn off at the end points. (Inset) Fourier transform of the bandwidth-limited shaking function.

Figure 7

Response of the reciprocal (red, solid) and nonreciprocal (blue, dashed) interferometers to a sinusoidal signal as in Eq. (6). The response is given in terms of the variation $D_{\mathrm{f},\mathrm{i}}$ between the optimized final state of the interferometer and the final state after shaking with the applied signal as in Eq. (4). The total interrogation time of each interferometer is 2.008 ms and the amplitude of the applied acceleration signal is $a_0=0.115$ m/${\mathrm{s}}^2$. The reciprocal interferometer is 20 times more sensitive than the nonreciprocal interferometer to a signal with $f\approx 7$ kHz, showing that SLI may be modified to tailor the interferometer response to an ac signal.

Figure 8

Response of an interferometer optimized in the presence of a dc bias acceleration $a_{\mathrm{dc}}=0.76$ m/${\mathrm{s}}^2$. The interferometer response is clearly minimized in the vicinity of $a_{\mathrm{dc}}$ and increases away from this bias. This shows that SLI can be used to reject a dc bias of a given magnitude or measure perturbations around this bias.

Figure 9

Minimum detectable acceleration $\delta a$ scaled by $\sqrt{N_{\mathrm{a}}}$ plotted on a log scale vs the interrogation time $T_{\mathrm{I}}$. The black points are simulation results, and the red line is a fit of the form $C{T}_{\mathrm{I}}^{-n}$. Here, lower values of $\delta a$ correspond to a more sensitive interferometer.

Figure 10

Acceleration sensitivity $\delta a$ (relative to $g$) for varying atom numbers $N_{\mathrm{a}}$ at $T_{\mathrm{I}}=10$ ms (red), 100 ms (blue), and 1 s (black).

Figure 11

Percent variation of the optimized splitting shaking function shown in Fig. 6 after variations of (a) the simulated lattice depth and (b) the simulated lattice wavelength. A variation of $1\%$ is marked in each plot with a blue dashed line.

Figure 12

Variation of the final state with varying noise amplitudes added to the optimized splitting shaking function. The results shown here are the average of five runs with random white Gaussian noise added to the shaking function $\varphi \left(t\right)$. Error bars give the standard deviation of the variations. Noise amplitude is given as a fraction of the maximum of the $\varphi \left(t\right)$ shown in Fig. 6. The red line marks a variation of $1\%$.

Figure 13

Variation of the final split state after the addition of spurious lattice potentials with varying phase due to unwanted reflections with reflection amplitudes of $\epsilon =0.1\%$ (red), $1\%$ (blue), and $4\%$ (cyan).