Using Time-Reversal Symmetry for Sensitive Incoherent Matter-Wave Sagnac Interferometry

We present a theory of the transmission of guided matter-waves through Sagnac interferometers. Interferometer configurations with only one input and one output port have a property similar to the phase rigidity observed in the transmission through Aharonov-Bohm interferometers in coherent mesoscopic electronics. This property enables their operation with incoherent matter-wave sources. High rotation sensitivity is predicted for high finesse configurations.

Figure 1

Geometries of guided matter-wave Sagnac interferometers: (a) MZ. Open squares mean controllable reflectivity. When fully (not) reflecting we name the interferometer: closed (open) MZ. (b) 2-port loop. Closed squares are fully reflecting mirrors. (c) 4-port loop. (d) Single junction loop. In all configurations horizontal lines mean 50%–50% BSs while vertical lines mean BSs with a transmission amplitude $i$t.

Figure 2

Transmissivity of Sagnac interferometers as a function of rotational phase shift $\varphi$ (a) closed MZ. (b) 2-port loop. (c) 4-port loop. (d) single junction loop. Thin curves in plots (a)–(c) represent the quasimonochromatic transmission $\overline{P}(\varphi )$ for few values of phase difference between the two arms $k\delta L$ and thick curves represent the transmission in the incoherent limit. The dashed curve in (a) represents the open MZ transmission in the incoherent limit. In (b)–(c) the BS transmission amplitude is $t=0.25$. Comparison of the closed and open MZ in (a), and comparison of (b) and (c), demonstrate how time-reversal symmetry $\overline{P}(\varphi )=\overline{P}(-\varphi )$ preserves the visibility when averaged over $k\delta L$ (incoherent limit). In (d) solid curves represent transmission $\overline{P}(\varphi )$ for few values of $t$ (numbers) and the dashed curve is the transmission for an incident thermal mixture of 101 transverse modes $j$ with $0<t_j<1$ (equal spacing) and $n_j^{(\mathrm{in})}\propto e^{-\kappa t_j}$ ($\kappa =10$). Transmission at small $\varphi$ is dominated by low $t_j$ states (high finesse). The dip in the incoherent signal of (b) and (d) at $\varphi =0$ is due to destructive interference between exact same length trajectories (due to $\pi /2$ phase difference between the $r$ and $t$ amplitudes).