Fundamental limitations of cavity-assisted atom interferometry

Atom interferometers employing optical cavities to enhance the beam splitter pulses promise significant advances in science and technology, notably for future gravitational wave detectors. Long cavities, on the scale of hundreds of meters, have been proposed in experiments aiming to observe gravitational waves with frequencies below 1 Hz, where laser interferometers, such as LIGO, have poor sensitivity. Alternatively, short cavities have also been proposed for enhancing the sensitivity of more portable atom interferometers. We explore the fundamental limitations of two-mirror cavities for atomic beam splitting, and establish upper bounds on the temperature of the atomic ensemble as a function of cavity length and three design parameters: the cavity $g$ factor, the bandwidth, and the optical suppression factor of the first and second order spatial modes. A lower bound to the cavity bandwidth is found which avoids elongation of the interaction time and maximizes power enhancement. An upper limit to cavity length is found for symmetric two-mirror cavities, restricting the practicality of long baseline detectors. For shorter cavities, an upper limit on the beam size was derived from the geometrical stability of the cavity. These findings aim to aid the design of current and future cavity-assisted atom interferometers.

Figure 1

An interferometer sequence using cavity-assisted large momentum beam splitters. A cloud of atoms is coherently split ($t=0$), deflected ($t=T$), and recombined ($t=2T$) using light pulses that resonate in the cavity.

Figure 2

Cavity-induced deformation of a Gaussian input. Envelope functions of the intracavity field for a 1 m cavity injected with a $1\mu \mathrm{s}$ pulse for different cavity finesses. All areas are normalized to the input pulse area for comparison. When the pulse duration is comparable to the photon lifetime of the cavity, its envelope function is elongated. Inset: Envelopes without normalization.

Figure 3

Transient response of the cavity to short pulses. Intracavity to input ratios of the pulse area (solid lines) and width (dashed lines) vs input pulse width, for four cavities of different length and finesse. The vertical lines represent the photon lifetime of each cavity.

Figure 4

Population of state $|+4\hslash k\rangle$ as a function of pulse width for a Gaussian envelope, for different values of the laser intensity. When the interaction time is short ($\delta t{\omega }_r⪅1$) the intermediate states cannot be adiabatically eliminated and the equations of motion need to be solved numerically considering all intermediate states and sufficiently many outer states.

Figure 5

Duration vs intensity of the first mirror pulses in the 10 m cavity for different values of the cavity finesse, indicated at the bottom of each curve. As the cavity finesse increases, the curves shift left as the beam splitters require less input power due to the increased optical gain (a). After reaching a particular value of the finesse ${\mathcal{F}}_{\max }$, the curves shift right and up, as the cavity-induced elongation becomes more severe and power enhancement of the beam splitters worsens (b).

Figure 6

Duration vs intensity of the first mirror pulses for varying cavity photon lifetimes. The required input power is minimized for ${\tau }_c={\tau }_{\max }$, in this case ($n=4$) ${\tau }_{\max }\approx 10\mu \mathrm{s}$. For ${\tau }_c>{\tau }_{\max }$, the minimum interaction time grows significantly. The input laser intensity scale shown here can be adjusted for any cavity length $L$ by applying a factor $L/1$ m.

Figure 7

First Rabi cycle of a conventional $2\hslash k$ beam splitter for three cavities with different photon lifetimes, indicated next to each curve. This transition is less lossy because of the direct coupling between the initial and final states. High-finesse cavities, with increased interaction times due to the severe elongation effect, exhibit a sharp transition between the adiabatic Bragg regime and the long-interaction steep-potential channeling regime, embodying the uncertainty relation between time and energy in the parameter space (a). The crosses represent the points where the transfer efficiency falls below 95% for the lower finesse cavities, as the interaction time approaches the Raman-Nath regime (b).

Figure 8

Effect of the cavity photon lifetime ${\tau }_c$ on the atomic transitions. The minimum photon-atom interaction time remains largely unaffected for cavities with ${\tau }_c<{\tau }_{\max }$, and increases linearly for ${\tau }_c>{\tau }_{\max }$ (a). The required intensity for the shortest beam splitter is minimal for ${\tau }_c={\tau }_{\max }$ (b).

Figure 9

Variation of ${\tau }_{\max }$ with the order of the diffraction process. The error bars represent the statistical uncertainty yielded by propagation of error through a least squares fit of the data to calculate the photon lifetimes that minimize the required input intensity of the beam splitters.

Figure 10

The bandwidth limit places a constraint on the cavity's $L\text{−}\mathcal{F}$ parameter space, depicted here for $n=1$. Higher finesses lead to better spatial filtering (a), while longer lengths allow for larger beams (b). In atom interferometry, both large beams and good spatial filtering are desired qualities.

Figure 11

Spatial filtering and geometrical properties of the cavity. The optical suppression factor of the first and second order spatial modes serve as indication of the quality of the laser wave fronts as a function of beam waist size (a). Cavities with the same bandwidth (16 kHz here, the limit for $n=1,4$) have the same spatial filtering properties in the large waist limit. Also plotted: variation of the total cavity $g$ factor (b), the roundtrip Gouy phase shift (c) and the mirror radii of curvature (d) of each cavity. As the beam waist varies from ${10}^{-2}$ to ${10}^2$ mm, the cavity geometries (e) go from near-unstable concentric ($\mathcal{R}\to L/2$, $g_{1,2}\to -1$, $\mathrm{\Delta }{\varphi }_G\to 2\pi$) through critically stable confocal ($\mathcal{R}\to L$, $g_{1,2}\to 0$, $\mathrm{\Delta }{\varphi }_G\to \pi$) and up to near-unstable plane-parallel ($\mathcal{R}\to +\infty$, $g_{1,2}\to +1$, $\mathrm{\Delta }{\varphi }_G\to 0$).

Figure 12

Geometrical and optical limits of the cavity-assisted atom interferometer. Beam waist size (b) and cloud temperature limits (a) are derived from a series of constraints. In the geometrical limit the upper bounds are set by the maximum cavity $g$ factor that is experimentally realizable (c). Longer cavities sit more comfortably within geometrical stability but in turn offer worse suppression of higher order spatial modes due to the bandwidth limit. In this region the upper bounds are set by the requirement of achieving a certain level of suppression of the first and second order spatial modes. The maximum cavity finesse is indicated in (d); cavities in the optical limit are by definition at the bandwidth limit, and thus also at the finesse limit. The temperature limits are calculated for two types of interferometric sequences (e). For on-axis sequences we show a case in which the atomic trajectories explore the entire cavity length and one in which they only use 1/10 of the length. For the perpendicular type we show the limits for total measurement times of 100 and 250 ms.