Sagnac Interferometry Using Bright Matter-Wave Solitons

We use an effective one-dimensional Gross-Pitaevskii equation to study bright matter-wave solitons held in a tightly confining toroidal trapping potential, in a rotating frame of reference, as they are split and recombined on narrow barrier potentials. In particular, we present an analytical and numerical analysis of the phase evolution of the solitons and delimit a velocity regime in which soliton Sagnac interferometry is possible, taking account of the effect of quantum uncertainty.

Figure 1

Stages of Sagnac interferometry. An incoming soliton splits at time $T_s$ on a barrier into two solitons of equal amplitude and opposite velocity. After circumnavigating the ring trap, at time $T_c$ the solitons recombine either at the same barrier (a), or a second barrier (b) antipodal to the first, illustrated in both cases with angular rotation $\mathrm{\Omega }=1.875\times {10}^{-3}$, and ring circumference $L=40\pi$. The resulting phase difference, incorporating the Sagnac phase due to the rotating reference frame, is read out via the population difference in the final output products within the positive (shaded) and negative domains. (c) Final population in the positive domain $I_{+}$ as a function of $\mathrm{\Omega }$, with $L=40\pi$ and initial soliton velocity $v=4$. The sensitivity of the single barrier case (dashed line) is twice that of the double barrier case (solid line) because the interrogation time $T_{\mathit{c}}-T_s$ is doubled.

Figure 2

Numerically calculated transmission into the positive domain after the second collision $I_{+}$, for the two Sagnac interferometry geometries shown in Fig. 1. Color maps for the (b) two barrier and (d) single barrier cases show the $0.16<v<4$, $0<\mathrm{\Omega }\times {10}^3<2.5$ parameter space. Panels (a) and (c) show specific curves of constant $v$ for these scenarios [for $v=0.52$ (dashed-dotted line), $v=1$ (dashed line), and $v=4$ (solid line)], and highlight how the different interrogation times result in a different Sagnac phase accumulation. The phase difference is varied by varying $\mathrm{\Omega }$ while keeping $L=40\pi$ [hence the $v$ ranged over in panels (b) and (d) correspond to dimensionless angular velocities of between $\omega =0.008$ and $\omega =0.2$].

Figure 3

Results of Monte Carlo simulations used to model effects of quantum uncertainty for a range of $v_0$, $N$, and $\mathrm{\Omega }$. (a)–(d)(i) Scatter plot of the solitons’ collisional velocity $v_b$ for ensembles of individual simulations. In (d)(i), the higher gradient of the curves through the points implies the detected transmission $I_{+}$ is less sensitive to quantum fluctuations. (a)–(d)(ii) Sample distributions of the simulation outcomes. For each $N,v_0$ pair we explored $\mathrm{\Omega }\times {10}^3=0$, 6.25, 12.50, 18.75, 25, corresponding to Sagnac phases of ${\delta }_S=0(+)$, $\pi /2(\circ )$, $\pi (▵)$, $3\pi /2(\square )$, $2\pi (\times )$, respectively. The simulations were carried out in a two barrier system, with $L=40\pi$ (hence the $v_0$ values correspond to dimensionless angular velocities $\omega =0.05$, 0.1). The $I_{+}$ peak locations are consistent with the GPE-predicted nonlinear skew for these velocities [18] (see also Fig. 2).