Controlling the dynamical scale factor in a trapped atom Sagnac Interferometer

Sagnac interferometers with massive particles promise unique advantages in achieving high precision measurements of rotation rates over their optical counterparts. Recent proposals and experiments are exploring non-ballistic Sagnac interferometers where trapped atoms are transported along a closed path. This is achieved by using superpositions of internal quantum states and their control with state-dependent potentials. We address emergent questions regarding the dynamical behavior of Bose-Einstein condensates in such an interferometer and its impact on rotation sensitivity. We investigate complex dependencies on atomic interactions as well as trap geometries, rotation rates, and speed of operation. We find that temporal transport profiles obtained from a simple optimization strategy for non-interacting particles remain surprisingly robust also in the presence of interactions over a large range of realistic parameters. High sensitivities can be achieved for short interrogation times far from the adiabatic regime. This highlights a route to building fast and robust guided ring Sagnac interferometers with fully trapped atoms.


I. INTRODUCTION
Atom interferometry [1] for precision measurements and quantum sensing [2] has become a powerful tool with applications ranging from fundamental physics [3] to absolute gravimetry [4] and inertial sensing [5,6]. If the effect to be measured depends on length or inertial and gravitational forces, the scaling of sensitivity with particle mass in an atom interferometer can be directly compared to its optical counterpart, promising signal gain by orders of magnitude [7]. In a Sagnac interferometer, the resulting phase φ S = 2 m Aω S can be used to measure rotation frequency ω S . This phase is proportional to the (equivalent) mass m and the area A enclosed by the interferometer, factors which combine into the scale factor ∂φ s /∂ω s . Despite much smaller particle flux and enclosed interferometer area, atom interferometric gyroscopes rival commercial fibre-optic devices. Sensitivities below 10 −9 rad/ √ s [8], [9] with thermal beams and stability below 10 −9 rad/s [10,11] with free-falling, lasercooled ensembles have been demonstrated, see [12] for a recent review. In order to reduce apparatus sizes and to gain operational independence from specific conditions of gravitation and acceleration, a range of ring-shaped atom traps and guided interferometers have been proposed and implemented with various means, geometries, and objectives [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Large enclosed areas are desired, and multiple cycles [14] or, equivalently, resonator approaches [15] have been proposed, although the scaling of decoherence due to longer path lengths may limit the possible benefit over physically large areas [16]. The majority of approaches relies on the ballistic motion of particles along a closed path, but the rotationdependent Sagnac phase can equally be measured with fully trapped atoms, i.e. confined in three dimensions [20,25,31]. In such a trapped Sagnac interferometer (TSI) atomic motion is actively controlled. The necessary beam splitting and recombination arise from coherent internal state operations in conjunction with state-dependent potentials. Fully trapped atoms promise some important advantages. Interference can be observed without a standing wave phase pattern, which may require high imaging resolution [24] and interferometric stability with respect to a reference that is external to the trap, e.g. camera position or a standing wave light field. Parti-cles can be accelerated to high speeds on path enclosing large areas within short times, a goal pursued by experiments on large momentum beam splitters [32][33][34]. Atomic wave packets do not disperse and their transport can be well controlled against gravity and external acceleration, where, in contrast, ballistic operation will affect the cycle time and may even preclude the enclosure of a large physical area.
Both guided and trapped interferometers have not yet reached maturity, and some intrinsic effects received little attention so far. The discussion in the context of quantum sensing is often focused on a quantum advantage that only affects shot-to-shot noise. But the advantage can easily become unsubstantial as many performance parameters must be considered that can be coarsely categorized as affecting precision and accuracy. E.g., only the combination of shot-to-shot phase noise, measurement bandwidth, and scale factor leads to meaningful short-term sensitivity, and these factors may be traded against each other. Uncertainties in the scale factor affect accuracy and stability. The intrinsic effects of guided matter-wave interferometers include excitation of higher trap modes by internal and external forces, such as centripetal forces and imperfections that alter particle trajectories [21,30], potential corrugations, external acceleration, and vibration. These will affect timing, enclosed area, and interferometer contrast, and understanding their impact is complicated further by atomic interactions, quantum degeneracy and dimensionality of the atomic ensemble. These effects alter the proportionality between the measured signal and rotation, i.e. the signal's scale factor, which is not simply given by the static factors that enter the expression for the Sagnac phase. An actual measurement is rather determined by a more involved, dynamical dependence of the interferometer output on external rotation, which defines a dynamical scale factor.
In this work, we investigate the dynamical scale factor in conjunction with measurement bandwidth for the case of a trapped two-mode Bose-Einstein condensate (BEC) and analyze a simple optimization scheme to achieve robust sensitivities, focusing on the slow rotation regime. Maximal sensitivities can be obtained when the spatial wave functions in the two interferometer arms remain identical. However, the transport will excite opposing center-of-mass motions in the trap, and these are not necessarily anti-symmetric between the two arms due to external rotation and atomic interactions. The path-dependent excitation severely reduces interferometer sensitivities. Through optimizing time-dependent driving profiles of the transport potential, we can robustly achieve near-maximal sensitivities at short interrogation times (hence large bandwidth) regardless of atomic interactions.

II. INTERFEROMETER MODEL
We consider an ensemble of N atoms with two internal states {|1 , |2 }, as depicted in Fig. 1a. These can be hyperfine levels of alkali atoms (e.g. Rb, Cs). Atoms can be put into coherent superposition of internal (clock) states and transported in opposite directions along a ring by state-dependent traps [13,25,35], which are guided along a ring with radius R p , as shown in Fig. 1a. For simplicity, we assume strong confinement in the direction perpendicular to the ring (z-axis) assuming that the system remains in the ground state in this direction. The dynamics of the BEC are governed by two coupled, two-dimensional (2D) Gross-Pitaevskii equations (GPEs) [36], where j, k = 1, 2 label the two internal states (components). We have used Ω jk to denote the pulse-driven coupling strength between the two states, with Ω jk = 0 if j = k. The order parameter ψ j (r, t) is normalized according to n 1 + n 2 = 1 with the j-th component occupation probability n j = |ψ j (r, 0)| 2 dr. The j-th component Hamiltonian is given by where m is the atomic mass and ω r (ω τ ) are the trapping frequencies in the radial (azimuthal) directions. In Eq. (1), we have defined local coordinate vectors The trap centers are determined by the driving function θ p (t) and the external rotation of angular frequency ω s that is to be measured. The boundary conditions for the driving function are θ p (0) = 0 and θ p (T ) = π. The coefficients g jk = 2 √ 2πN 2 a jk /(ml z ) quantify the strengths of intra-state (j = k) and inter-state (j = k) interactions, which depend on the number of particles N , and the effective s-wave scattering lengths a jk under the outof-plane confinement length l z [37]. For convenience, we scale time, energy and length according to t s = 1/ω r , E s = ω r and l s = /2mω r in the following unless stated explicitly.

III. SENSITIVITY OF THE SAGNAC INTERFEROMETER
To operate the TSI, we first create a coherent superposition state (|1 + |2 )/ √ 2 by performing a fast π/2-pulse on a BEC in the internal state |1 . Here and in the following, we neglect dynamics of the spin rotation process as Ω 12 (Ω 21 ) is on the order of MHz, which is far larger than typical energy scale (kHz) of the BECs. To take account of atomic interactions, the ground state and dynamics of the trapped BECs are obtained by numerically solving Eq. (1) with an imaginary and real-time algorithm, respectively. The two states are then guided along a ring in opposite directions (see Fig. 1a). When they are recombined after the interrogation time T [12], a second fast π/2-pulse is applied to convert the accumulated phase difference into a population difference between the two states. The rotation frequency ω s is then encoded in the expectation value σ z = n 2 − n 1 . The detection can be sensitive, if the differential response of population difference to rotation rate is large. At the same time, noise should be low and the interrogation time T should be short, equivalent to high measurement bandwidth (∼ 1/T ). Combining these considerations, the figure of merit is given by the interferometer's short term sensitivity S, which equals angular random walk of the time integrated rotation estimate. Assuming shot noise of uncorrelated particles near balanced output, it is given by which depends on the dynamical scale factor Σ(ω s ) and improves for increased particle throughputṄ = N/T , i.e. the number of particles per cycle N and the cycle which measures changes of the expected signal σ z with respect to angular rotation frequency ω s . Here, we used the parameterization σ z = ρ sin φ, where ρ and φ describe length (contrast) and orientation (phase) of the Bloch vector, see Fig. 1b. Both quantities are determined by the spatial overlap of the two states ψ * 2 (r, T )ψ 1 (r, T )dr = ρe iφ [20] (see Appendix A for details). Imperfect spatial overlap occurs if the centerof-mass positions or momenta of the two states do not coincide at the end of the sequence even if their wave functions share similar shapes [39,40]. More importantly, atomic interactions and external rotation together can significantly distort wave functions of the two states asymmetrically, which significantly reduces the overlap. In the following, we will focus on influences of the latter on the sensitivity.
In the slow rotation limit ω s → 0, the derivative ∂ρ/∂ω s of the scale factor Eq. (4) vanishes because ρ must be an even function of ω s . Here, the maximum scale factor (obtained by setting the phase reference such that φ = 0 at ω s = 0) reduces tõ Σ = Σ(ω s = 0) = ρ ∂φ ∂ω s ωs=0 , which solely depends on ρ and the phase gradient ∂φ/∂ω s . In the following we will investigate how these two parameters depend on the atomic interaction, the trap aspect ratio, and the interrogation time T . First, we consider a simple linear driving profile θ p (t) = tπ/T and a non-interacting BEC. Fig. 2a shows the scale factor obtained by numerically solving the coupled GPEs with a small rotation (we take ω s = 10 −3 ω r and R p = 10, throughout the work). The scale factor oscillates as a function of the interrogation time T , with decreasing amplitude for increasing T [20]. The reason is that intrap oscillations excited by the initial acceleration will be stopped by the final deceleration, if the driving profile does not contain the corresponding spectral components [20]. The oscillation frequency is thus the same as the trapping frequency ω r , which signifies the elementary excitations in the moving trap that affects the interference of the two matter-waves. In the limit T → ∞, the scale factor is approximately given bỹ It thus approaches the conventional scale factor 2A ∞ , where A ∞ = πR 2 p is the area of the ring (in scaled units of l 2 s ). The denominator accounts for the correction of the centrifugal effect (see Appendix C). This dependence provides a first example for a dynamical scale factor: the centrifugal forces lead to an increase in the area enclosed by the atomic trajectories. This effect is present in higher dimensions (2D and 3D) whilst absent in 1D models.
Scale factors change qualitatively when inter-and intra-state interactions are taken into account. We observe that the oscillations of the dynamical scale factor increase drastically at intermediate interrogation times, and their amplitudes decrease much slower with increasing T (Fig. 2a) than in the non-interacting case. The reduction of scale factors arises from the fact that collective modes of the BEC are excited when atoms are transported non-adiabatically [41] around the ring. To illustrate this, we evaluate the projected BEC densities on thẽ τ axis, I j = dr|ψ j | 2 right before and right after the second π/2-pulse. As shown in Fig. 3a for a non-interacting BEC, the densities of the individual components may be shifted oppositely from the trap center before the second pulse, leading to incomplete conversion of phase into population difference. In addition, the density profiles change with increasing interaction strengths (Fig. 3bd). When intra-state interactions dominate (Fig. 3d), the wave packets distort significantly from a Gaussian shape during the transport.

IV. OPTIMIZATION FOR A NON-INTERACTING AND INTERACTING BEC
In the following we aim to avoid the path dependent excitation of the BEC components in order to reach maximal scale factorΣ ∞ as well as short interrogation times. Previous studies [20,31] have considered ideal driving functions of a non-interacting BEC by excluding frequency components at the trapping frequency, T 0 sin[θ p (τ )]e iτ dτ = 0. However, this condition does not avoid oscillations during the transport and it is insufficient when interactions are non-negligible.
During the transport, the BEC should be accelerated (decelerated) slowly at t → 0 (t → T ) to avoid dynamical excitations, which is satisfied by the nonlinear driving function where the (Beta-)function B(a, b) ensures normalization, to meet the boundary condition θ p (T ) = π. This driving function is a convenient choice and has been applied to other optimization problems, e.g., conformal antenna arrays [42]. It generally has a sigmoidal form and includes the linear ramp as a limiting case, as shown in Fig. 2b. Using this driving function, the dynamical scale factor for a non-interacting BEC is shown in Fig. 2a as a function of the interrogation time T . The oscillations have vanished and the scale factor quickly approaches the static valueΣ ∞ already at short times. The improvement of sensitivity and robustness is rooted in the fact that dynamical excitations of the BEC are suppressed significantly as the driving function has a vanishing slope at the beginning and end of the interferometer operation (see Fig. 2b), which ensures the BEC is accelerated and decelerated slowly. Note, that this behavior is largely independent of precise choices of the parameters a and b, which can be seen in Fig. 2d.
To study a TSI implemented with an interacting BEC, we first consider a scenario where atomic interactions are symmetric (g 11 = g 22 = g 12 = g) and the trapping potentials are isotropic (ω r = ω τ ). We use the same driving function θ p (t) with parameters a = b = 4 as in the case of a non-interacting BEC. In Fig. 4a, the resulting dynamical scale factors are presented for various interaction strengths g. Although oscillations of the scale factor reemerge with stronger interactions, their amplitudes quickly decrease with increasing T . For the strongest interactions considered in these examples (g = 30), the dynamical scale factor settles near the maximal valueΣ ∞ for T > 100.
Similar observations hold in the case of anisotropic trapping potentials (Fig. 4c,d) as well as for nonsymmetric atomic interactions (Fig. 4e,f). To illustrate effects due to trap anisotropy, we consider an example where ω r = 2ω τ while the interactions are still symmetric. Due to the strong radial trapping, one expects a weaker centrifugal force. As a result, the maximal value of the scale factor slightly decreases at small T for noninteracting BECs, as shown in (Fig. 4c). In the presence of two-body interactions, dynamical scale factors oscillate with larger amplitudes in the small T region. Increasing T , the optimized scale factor quickly approaches the maximal value. When the inter-and intra-state interactions differ, i.e. for g 11 = g 22 = g = g 12 , we find that robust dynamical scale factors can be obtained when g 12 is smaller or comparable to g. However, the optimized (non-interacting) driving function becomes less efficient when g 12 is much greater than g. We attribute this to the fact that strong repulsion [41] between the two BEC components causes immiscibility and prevents them from overlapping in space (see Fig. 3d), leading to reduced contrast. This, however, is not a major issue in realistic experiments as the inter-and intra-state scattering lengths can be very similar (e.g. Rb atoms). An interesting observation is that in all the considered cases, the scale factor is mostly influenced by a reduction of contrast rather than through the phase gradient. This can be seen by the very similar dependencies of scale factor and contrast on the interrogation time T , as shown in the insets to Fig. 2a,b,c. The scale factor exhibits a weak dependence on the phase gradient at intermediate interrogation times 20 < T < 50 (see Fig. 4b,d,f), where values going beyond the static scale factor are achieved due to non-negligible centrifugal forces. An important finding is that the phase gradient is largely immune to atomic in-teractions and trap geometry, which we attribute to the suppression of radial center-of-mass oscillations also in the presence of interactions. As shown in Fig. 4b,d,f, and similar to the optimized response in Fig. 2d, the phase gradient decreases smoothly with increasing T , approaching the static scale factor for T → ∞. In fact, the scale factor agrees closely with numerical calculations of the path enclosed area for a point-like classical particle (see Appendix C for details), which is equivalent to trajectories of the center-of-mass of the BEC [43,44]. We find a good agreement even when we weigh the point-wise particle's position with the BEC wave packet. Therefore, an accurate measurement of rotation can be obtained by adaptive phase estimation protocols that co-estimate the contrast also in the case of uncertain dynamics [45,46].
Finally, we present some experimental parameters relevant to this study. Using 87 Rb as an example, timeaveraged-adiabatic-potential traps [25] can confine the atoms in radial direction at frequency ω r ≈ 2π × 127 Hz, and ω z ≈ 2π × 206 Hz in z-direction. This results in the unit time, length and energy to be t s = 1.26 ms, l s = 1.70 µm, and E s = 8.35 × 10 −32 J respectively. The scattering length of 87 Rb is nearly identical in |F = 1 and |F = 2 states for either intra-state or inter-state interactions. The difference is below 1% and the average value isā s ≈ 98a 0 , where a 0 is Bohr radius [47]. In the 2D model, the effective interaction strength is g ij = N mω z /2π (4π 2 /m)a ij where N is the number of atoms [37]. Using E s and l s , we obtain g = 4.39 × 10 −3 N E s l 2 s . To change the relative interaction strength, we can, e.g., vary the number of atoms in the BEC. For example, the dimensionless interaction strength will be g = 43.9 for 10 4 atoms. Tuning the interaction strength will allow for exploration of the effects predicted by this investigation.

V. CONCLUSION
We analyzed the dependence of the dynamical scalefactor of guided Sagnac interferometers in the slow rotation regime with respect to transport parameters, atomic interactions, and trap symmetries. Employing a simple optimized driving function for the transport of atoms in state-dependent potentials reduces path-dependent excitations and achieves maximal sensitivities at moderate interrogation times, typically tens of trap oscillation periods, for both ideal and interacting BECs. Our theoretical study is important to guide current experimental efforts on building robust and fast Sagnac interferometers with fully trapped atomic gases. It lays a foundation for further analysis of other experimentally relevant parameters, such as atom number fluctuations [48,49], imperfect state operations, and finite temperatures [20].