Chip-Scale Point-Source Sagnac Interferometer by Phase-Space Squeezing

Abstract

Matter-wave interferometry plays a significant role in scientific research and technological applications. While position-momentum phase-space squeezing has been demonstrated to increase the coherence of atom sources by reducing momentum spread, we theoretically investigate the potential advantages of the opposite squeezing. As a case study, we analytically and numerically examine its effect on point source atom interferometry (PSI) for rotation sensing. Our analysis reveals that this squeezed PSI (SPSI) approach can significantly improve sensitivity and dynamic range while enabling shorter cycle times and higher repetition rates. Through simulations, we identify parameter spaces where sensitivity and dynamic range are enhanced by orders of magnitude. Under a specific definition of compactness, our calculations show that SPSI outperforms standard PSI by over four orders of magnitude. These theoretical findings suggest that SPSI could either enhance performance in standard-sized devices or maintain performance in miniaturized chip-scale devices, potentially paving the way for new practical applications.

Figures (5)

• Figure 1: Principle of the squeezing effect in the interferometer. The atomic phase-space distribution is shown for the standard PSI interferometer (blue) and the squeezed-PSI (SPSI) interferometer (orange), at two different times: (A) at the start of the interferometer sequence, $t_{\rm{acc}}$, occurring just after the acceleration stage due to the repulsive-potential pulse, and (B) at the imaging time, $t_\text{final}$, following the complete interferometer sequence. The phase-space distribution at $t_{\rm{final}}$ becomes tilted due to expansion, and the velocity-dependent interferometer phase introduces an oscillatory phase $\Delta\phi(v_x)=2k_{\rm eff}\Omega_y T_R^2 v_x\equiv k_v v_x$ that modulates the phase-space distribution as $\rho(x,v_x)\to \frac{1}{2} \rho(x,v_x)[1+\cos(k_v v_x)]$ upon detecting a single state. As shown at the bottom of (B), the latter oscillation produces a spatial density oscillation, which, upon being projected onto the position axis, forms the observed signal (output) of the interferometer. In the SPSI, when the initial phase-space distribution is squeezed with increased velocity, the subsequent expansion results in a phase-space distribution having a high aspect ratio. Upon projection onto the position axis, the SPSI signal shows improvement with more oscillations and an improved contrast. This figure was generated using a simplified version of the numerical simulations described in the Materials and Methods section, focusing on illustrating the key principles of phase-space evolution in both PSI and SPSI configurations.
• Figure 2: Contrast and dynamic range. (A) Contrast vs. angular velocity ($\Omega$). Using $^{87}$Rb atoms, the simulation parameters are initial cloud size $\sigma_{x0} = 100\,\rm{\mu m}$, temperature $T = 5\,\rm{\mu K}$, and time between pulses $T_R = 5\,\rm ms$. For the repulsive potential, we take a beam power of $P=1\,$W with a cross-section of $400 \cross 400\,\mu$m$^2$ (according to the beam profile of Eq. \ref{['eq:harmonic potential frequency']}), blue-detuned by $\Delta = 2\pi\cdot10\,$GHz, and focused such that in the direction of acceleration it gives rise to an inverted harmonic potential and in the transverse direction a constant potential. The harmonic profile was optimized to achieve a large squeezing factor before the atoms exceed the region where the repulsive potential is harmonic. Following Eqs. \ref{['eq: squeezing parameter']} and \ref{['eq:harmonic potential frequency']}, and choosing $t_\text{acc} = 80.6\,\mu$s, we find a squeezing parameter of 22.81. The purple line represents the analytical solution of PSI contrast, while the dashed orange line corresponds to SPSI, according to Eq. \ref{['eq: Gaussian cloud contrast']}. The simulation points are the result of a numerical simulation. While the contrast reduction due to the ratio between the fringe periodicity and the initial cloud size determines the upper limit of the detection range, there exists a lower limit when the fringe period becomes larger than the final cloud size. These limits are roughly given by $\Omega_{\rm min}$ and $\Omega_{\rm max}$ (Eq. \ref{['eq: PSI limits']}), which are presented in the graph as vertical dashed lines. It is evident that using the SPSI greatly improves the contrast, thereby increasing the detection dynamic range. At high angular velocities, the contrast obtained from simulation decreases faster than the analytical solution due to the short spacing between fringes relative to the detection pixel size considered only in the numerical calculation. (B) The analytical curve of the detection range as a function of $\eta$ according to Eq. \ref{['eq: PSI limits']}. The dashed line presents the SPSI simulated in (A) with $\eta = 22.81$.
• Figure 3: Relative single-shot sensitivity and measured angular velocity deviation. (A) Relative sensitivity $\delta \Omega_\text{single} / \Omega$ vs $\Omega$, where $\Omega$ is the nominal angular velocity, for the same simulation parameters and notation as in Fig. \ref{['fig:Contract and dynamic range']}. The analytical curves are determined according to Eq. \ref{['eq: PSI sensitivity']}. The simulation results for PSI at large angular velocities where $\delta\Omega_\text{single}/\Omega>1$ are irrelevant and therefore not shown. (B) Simulation results of the measured angular velocity deviation $\Delta \Omega$ (Eq. \ref{['eq: Gaussian cloud deviation']}) for different angular velocities. Here $\Delta \Omega$ the discrepancy between the measured and nominal angular velocities, while the sensitivity (uncertainty) $\delta\Omega_\text{single}$ is represented by the error bars. $\Omega$ is extracted from each simulation run and obtained from the fringe spatial frequency of the fitted sinusoidal pattern (as described in Fig. \ref{['fig:Simulation Example']}). The data point labels are the same as in (A), while the black dashed line illustrates the point-source (PS) limit of zero deviation. The simulation results for PSI are not shown for large angular velocities where $\delta\Omega_{\rm single} / \Omega > 1$. The enhanced performance of the SPSI in dynamic range and sensitivity (measurement uncertainty) is clearly visible.
• Figure 4: Sensitivity-compactness trade-off in the SPSI. The sensitivity ratio $\delta\Omega_{SPSI}/\delta\Omega_{PSI}$ based on Eq. \ref{['eq: SPSI sensitivity']}, is colour coded against the squeezing parameter $\eta$ and the cycle time ratio $a_t = T_{SPSI} / T_{PSI}$ for an angular velocity of $\Omega = 100\,\rm{mrad/s}$, while maintaining constant values for the remaining parameters: Time between pulses $T_R = 10\,\rm{ms}$, temperature $T = 2\,\rm{\mu K}$, and initial cloud size $\sigma_{x0} = 100\,\rm{\mu m}$. The maximum value of $\eta$ was chosen to ensure that the majority of atoms are affected by the same potential throughout the acceleration phase. The bold black line indicates the validity boundary of this plot beyond which the minimal detectable angular velocity of the SPSI method (Eq. \ref{['eq: dynamic range - short time']}) exceeds the nominal angular velocity $\Omega$. It is evident that the implementation of a repulsive potential allows for a sensitivity enhancement of more than an order of magnitude while simultaneously reducing the cycle time by a factor of about two (as indicated by the bright red colour) or reducing the cycle time by approximately tenfold without affecting the sensitivity (as indicated by the light-blue colour). Specifically, utilizing the compactness factor definition of Eq. \ref{['eq: compactness factor']}, we find that with $\eta=40$ and $a_t=0.1$, we have a performance enhancement of $\kappa_{SPSI}/\kappa_{PSI} \simeq 2\cdot 10^4$ (black point).
• Figure 5: Simulation fringe pattern fit. This figure compares the simulation results of both methods, the top for PSI and the bottom for SPSI. The light blue shaded area depicts the total number of atoms in each bin, denoted as $N_{\text{total}}$, while the darker shade indicates only those atoms in the internal state $|1\rangle$, denoted as $N_{|1\rangle}$. The green points represent the normalized probability for the $|1\rangle$ state, calculated as $P_1 = N_{|1\rangle}(x)/N_{\text{total}}(x)$, while the red line denotes the sinusoidal fit to these green points. The angular velocity is determined from the fit's fringe spatial frequency using Eq. \ref{['eq: Gaussian cloud output density']}. In this example the initial 1D Gaussian cloud comprising $N=10^6$ atoms, with a width of $\sigma_{x0}=100\,\rm{\mu m}$ and a temperature of $T=5\,\rm{\mu K}$, as in Figs. \ref{['fig:Contract and dynamic range']} and \ref{['fig:Relative-sensitivity and Delta omega']}. For the PSI method's simulation, the atoms undergo free expansion during the interferometer sequence, with a time interval between pulses of $T_R = 25\,\rm ms$ used in this figure, while the simulated angular velocity is $\Omega = 40\,\rm{mrad/s}$. In the case of SPSI, the cloud is additionally subjected to a repulsive potential having a power of $P = 1\,\rm{W}$, with a beam cross-section of $400 \cross 400\,\mu$m$^2$, as in Figs. \ref{['fig:Contract and dynamic range']} and \ref{['fig:Relative-sensitivity and Delta omega']}, activated for $t_{\text{acc}} = 20\,\rm{\mu s}$. This results in a potential frequency of $\omega = 2\pi\times 1.214\,\rm{kHz}$ (Eq. \ref{['eq:harmonic potential frequency']}), a squeezing parameter $\eta = 5.4$ (Eq. \ref{['eq: squeezing parameter']}), and an effective time $t_{\text{eff}} = 0.84\,\rm{ms}$ (Eq. \ref{['eq: effective time']}). Consequently, the cloud size increases notably, and we chose the potential parameters here such that it would be easy to compare to the PSI simulation. Note that the contrast and number of fringes in the PSI method is smaller than that of the SPSI (see Figs. \ref{['fig:Phase-Space Squeezing']}-\ref{['fig:Contract and dynamic range']}), leading to a fit deviation and uncertainty one order of magnitude higher for the PSI case. It is important to note that while the histograms appear different in scale, the total atom count remains constant for both methods. The discrepancy in appearance is due to different bin sizes used in the PSI and SPSI plots to accommodate the varying cloud expansions, with SPSI utilizing larger bins to cover its greater spatial extent.