Spin-orbit coupled interferometry with ring-trapped Bose--Einstein condensates

TL;DR

This paper introduces a novel atom-interferometry method using spinor Bose-Einstein condensates with a magnetic field to create long-lived superpositional states, enhancing sensitivity to rotations and magnetic fields.

Contribution

It presents a new protocol for spin-orbit coupled interferometry in ring-trapped BECs that is deterministic, does not require optical coupling, and maximizes sensitivity for uncorrelated particles.

Findings

Protocol creates long-lived superpositional counterflow states.

Maximizes classical Fisher information for various sensitivities.

Sensitivity improves with longer interrogation times, limited by condensate lifetime.

Abstract

We propose a method of atom-interferometry using a spinor Bose-Einstein condensate (BEC) with a time-varying magnetic field acting as a coherent beam-splitter. Our protocol creates long-lived superpositional counterflow states, which are of fundamental interest and can be made sensitive to both the Sagnac effect and magnetic fields on the sub-micro-G scale. We split a ring-trapped condensate, initially in the $m_f=0$ hyperfine state, into superpositions of internal $m_f=\pm1$ states and condensate superflow, which are spin-orbit coupled. After interrogation, relative phase accumulation can be inferred from a population transfer to the $m_f=\pm1$ states. The counterflow generation protocol is adiabatically deterministic and does not rely on coupling to additional optical fields or mechanical stirring techniques. Our protocol can maximise the classical Fisher information for any rotation, magnetic field, or interrigation time, and so has the maximum sensitivity available to uncorrelated particles. Precision can increase with the interrogation time, and so is limited only by the lifetime of the condensate.

Equations

$$\begin{gathered} i\hslash \frac{\partial }{\partial t}{\Psi }_j=\left[-\frac{{\hslash }^2}{2m}{\nabla }^2+V-i\hslash (\mathrm{r}\times \mathbf{\Omega })\cdot \nabla +g_{\mathrm{n}}{\mathbf{\Psi }}^{†}\mathbf{\Psi }\right]{\Psi }_j \\ +{\left\{\left[g_{\mathrm{s}}\bar{\mathbf{F}}\cdot \mathbf{F}-{\mu }_{\mathrm{B}}{g}_F\mathbf{B}\cdot \mathbf{F}\right]\mathbf{\Psi }\right\}}_j, \end{gathered}$$

$$\mathbf{B}\cdot \mathbf{F}=\left(\begin{array}{ccc}{B}_z& B_{\mathrm{q}}{\mathrm{e}}^{i\varphi }/\sqrt{2}& 0\\ B_{\mathrm{q}}{\mathrm{e}}^{-i\varphi }/\sqrt{2}& 0& B_{\mathrm{q}}{\mathrm{e}}^{i\varphi }/\sqrt{2}\\ 0& B_{\mathrm{q}}{\mathrm{e}}^{-i\varphi }/\sqrt{2}& -B_z\end{array}\right).$$

$$|\pm B⟩=$$

$$|Z⟩=$$

$$|\Psi ⟩$$

$$=\sqrt{\frac{1+\cos (\delta )}{2}}|Z⟩-i\sqrt{\frac{1-\cos (\delta )}{2}}\left(\frac{|+B⟩+|-B⟩}{\sqrt{2}}\right).$$

Full text

The endeavour to optimally apply matter-wave interferometry has generated many proposals and prototypes for ultra-sensitive rotational Lenef et al. (1997); Gustavson et al. (1997); Halkyard et al. (2010); Helm et al. (2015); Moxley et al. (2016); Nolan et al. (2016), gravitational or inertial Snadden et al. (1998); Peters et al. (2001); McGuirk et al. (2002); Chu et al. (1999); Müller et al. (2008, 2010); Altin et al. (2013); Canuel et al. (2006), and gravity wave Tino and Vetrano (2007); Dimopoulos et al. (2008); Graham et al. (2016) detection protocols. In parallel, optical confinement potentials allow simultaneous trapping of atoms in different magnetic sublevels, constituting a spinor condensate Ho (1998); Ohmi and Machida (1998); Stamper-Kurn and Ueda (2013); Kawaguchi and Ueda (2012). In addition to their coherent nature, the ability to precisely manipulate motional and spin degrees of freedom using optical, radio-frequency, and magnetic fields makes spinor condensates a good candidate for the construction of an interferometer. We focus on a common path interferometric protocol, applying it to rotational sensing via Sagnac interferometry (where we note that our general common-path method is also applicable to zero-area Sagnac interferometry (ZASI) Sun et al. (1996); Eberle et al. (2010), an often discussed alternative to the Michelson geometry for optical gravity-wave detection Mizuno et al. (1997); Punturo et al. (2010); Mavalvala et al. (2010); Abbott (2016)).

In this Letter, we propose a method of matter-wave interferometry in which a repulsively interacting spinor BEC is split into a superpositional counterflow state Thanvanthri et al. (2008); Kapale and Dowling (2005) through the use of topological vortex imprinting Isoshima et al. (2000); Ogawa et al. (2002); Xu et al. (2010); Lovegrove et al. (2016); Stamper-Kurn and Ueda (2013); Kawaguchi and Ueda (2012), where the texture of an externally applied time-varying magnetic field ($B$-field) is embedded in the condensate’s spin and, hence, its phase. In the counterflow state, each atom is in a superposition of both spin and superflow, simultaneously moving clockwise and counter-clockwise, while also occupying multiple hyperfine sub-levels. This class of states is also of fundamental interest in that it yields superfluid–superfluid counterflow where the complicating effects of density gradients are substantially reduced Hamner et al. (2011). As the spin and angular-momentum degrees of freedom are linked, they can be said to be spin-orbit coupled, and we refer to the method as spin-orbit coupled interferometry (SOCI). This method is comparable to that proposed by Halkyard et al.  Halkyard et al. (2010) and has similarly maximised classical Fisher information (denoted $F_{\mathrm{C}}$) (Haine, 2016). The procedure uses experimentally accessible time-varying $B$-fields as a “beam-splitter” [Fig. 1]. The “arms” of the interferometer are not spatially separate, constituting a common-path interferometer insensitive to a variety of perturbing factors due to its intrinsic symmetry. Advantages of our SOCI method are that: (1) our interferometer can maximize $F_{\mathrm{C}}$, and so is shown to have the highest sensitivity achievable in the absence of entanglement; (2) the symmetry and common-path geometry preclude many systematic phase aberrations; (3) only standard magnetic fields are required for beam-splitting, with no optical phase-imprinting Thanvanthri et al. (2008); Kapale and Dowling (2005), mechanical stirring or weak link Ragole and Taylor (2016); Eckel et al. (2014); Jendrzejewski et al. (2014) required; (4) the precision of our interferometer is limited only by the lifetime of the condensate.

We present our SOCI method analytically in the context of an idealized measurement, and show that it converts an accumulated phase-difference (due to rotation at angular frequency $\Omega$, for example) between the counterflowing components into a difference among the populations, $n_{i}$, of the spin states. The measurement sensitivity of such a method can quantified by the classical Fisher information, $F_{\mathrm{C}}=\sum_{i}(\partial_{\Omega}n_{i})^{2}/n_{i}$ (Haine, 2016). Using fully-3D numerical simulations of the spinor mean field, for experimentally realistic parameters, we show that our method maximises $F_{\mathrm{C}}$ in the sense that $F_{\mathrm{C}}$ can be made equal to the maximum quantum Fisher information, $F_{\mathrm{Q}}$, achievable for uncorrelated particles (Haine, 2016).

We treat a spin-$F$ condensate as a system of $2F+1$ coupled BECs, working only in the $z$-quantised (ZQ) representation, where all spin states are labelled with reference to the $z$-axis. The vector-valued order parameter is ${\bf\Psi}=\sum_{j=-1}^{1}\Psi_{j}|j\rangle$, where $F_{z}|j\rangle=j|j\rangle$, and $\hbar F_{z}/2$ is the $z$-direction angular momentum operator in the ZQ basis. To consider the effect of rotations, we introduce an angular momentum term Aftalion and Du (2001); Mason and Aftalion (2011) characterised by the angular velocity vector $\bf{\Omega}$; our full mean-field dynamical equations are then Stamper-Kurn and Ueda (2013); Kawaguchi and Ueda (2012)

[TABLE]

where the local spin vector $\mathbf{\bar{F}}$ has components ${\bar{F}_{\alpha}}=\Sigma_{j,k}\Psi_{k}^{*}\Psi_{j}\langle k|F_{\alpha}|j\rangle$. Here we have atomic mass $m$, Bohr magneton $\mu_{\mathrm{B}}$ and hyperfine gyromagnetic ratio $g_{F}$ ($=-1/2$ for 87Rb in the $F=1$ manifold). The scattering terms are the normal interaction strength $g_{\mathrm{n}}$ and spin-spin interaction strength $g_{\mathrm{s}}$ Stamper-Kurn and Ueda (2013). The $V=m\omega_{\perp}^{2}[(\rho-R_{0})^{2}+z^{2}]/2$ term describes an optical ring trap Bell et al. (2016), where $\rho=\sqrt{x^{2}+y^{2}}$, giving a radial trapping frequency $\omega_{\perp}$ and major radius $R_{0}$. While we restrict our analysis to this specfic potential we note that a more general toroidal potential (with density zero at $\rho=0$) could be used to realize a similar interferometer. Gravity is taken to act in the $z$ direction and does not alter the symmetry, and so we do not consider it further. In our numerics we consider experimental parameters comparable to those described in Rakonjac et al. (2016); Ray et al. (2014), however, for faster numerics, we take the radial trapping frequency to be $\omega_{\perp}=2\pi\times 80$ Hz, the major radius of the ring to be $R_{0}=5a_{\perp}=6.02~{}\mu$m, and the number of 87Rb atoms to be $N=10^{4}$.

The idealized behavior of the system can be understood through the eigenvectors of the ${\bf B}\cdot{\bf F}$ operator, in turn determined by the texture of the magnetic field. Our fundamental requirement is that the $B$-field should have a non-trivial topology, such that a curve encircling the origin has non-zero winding number, which is satisfied by either an anti-Helmholtz or Ioffe-Pritchard (IP) coil configuration. We consider the geometrically simpler IP configuration, which is quadrupolar in the $x$–$y$ plane. The Cartesian components can then be written using cylindrical coordinates $\{\rho,\phi,z\}$ as ${\bf B}_{{\mathrm{I}\mathrm{P}}}=({B}_{\mathrm{q}}(\rho)\mathrm{cos}(\phi),-{B}_{\mathrm{q}}(\rho)\mathrm{sin}(\phi),{B}_{z})$, where the quadrupolar field ${B}_{\mathrm{q}}(\rho)=b^{\prime}\rho$ varies linearly with $\rho$ and the $z$ bias field is spatially uniform. For $F=1$, in matrix representation $|1\rangle=(1,0,0)^{T}$, $|0\rangle=(0,1,0)^{T}$, and $|-1\rangle=(0,0,1)^{T}$:

[TABLE]

The (spatially dependent) eigenvectors of Eq. (2) are:

[TABLE]

where $B=({B}_{\mathrm{q}}^{2}+{B}_{z}^{2})^{1/2}$. The $|+B\rangle$ and $|-B\rangle$ eigenvectors denote the strong- and weak-field-seeking states with eigenvalues $\pm B$, while $|Z\rangle$ is field-insensitive with eigenvalue [math]. Through these eigenvectors we can see the imprinting technique of Ref. Isoshima et al. (2000); varying ${B}_{\mathrm{q}}$ (via $b^{\prime}$) and ${B}_{z}$ over time, the condensate remains in a given eigenvector of ${\bf B}\cdot{\bf F}$, but transfers between the $m_{f}$ states, accumulating $l=F=1$ quantum of angular momentum. Some radial dynamics can occur as the $B$-field evolves, but these analytically separate out from the behaviour described by $\mathbf{B}\cdot\mathbf{F}$ Isoshima et al. (2000), and so are not addressed by our analytics. This implication of spin-gauge symmetry is confirmed by the full 3D numerics.

To achieve the counterflow state we must first prepare our condensate in the $|0\rangle$ spin-state with a large $z$ bias field $|{B}_{z}|\gg|{B}_{\mathrm{q}}(R_{0})|$. This constitutes the $|Z\rangle$ state. The $|0\rangle$ initial state can be achieved through RF-pumping a $|-1\rangle$ (weak-field-seeking) condensate Chang et al. (2004), following transfer to an optical trap, where magnetic trapping is no longer required. With the initial condition fixed in the $|Z\rangle$ state, we obtain the counterflow state by ramping $|{B}_{z}|$ down to zero over a period $T_{\mathrm{s}}$ [Fig. 1], splitting the condensate into a superposition of spin up and spin down [see  Eq. (4)]. We numerically explore two parameter regimes: (1) the quadrupolar field is characterised by an initial gradient $b^{\prime}=3.7$ G/cm while the initial $z$ bias field is set to ${B}_{z}=50$ mG [Fig. 1] (these parameters are consistent with Ref. Ray et al. (2014)); (2) we increase the field strengths by a factor of ten to separate the Zeeman and nonlinear timescales, producing a smoother response curve (Fig. 2). We select the ramp-down period $T_{\mathrm{s}}=32$ ms (or $3.2$ ms for the stronger $B$-field numerics) to be fifty times the Larmor precession time $T_{\mathrm{L}}=2\pi\hbar/(\mu_{\mathrm{B}}g_{F}b^{\prime}R_{0})=0.64$ ms ($0.064$ ms for the stronger $B$-field numerics), ensuring the spins follow the $B$-field adiabatically. The stronger fields are generally easier to generate experimentally and are easier to vary adiabatically due to their faster associated timescale, putting less stringent requirements on the level of field control. As the atoms in a given spin state have an associated flow field, the condensate is now in a superpositional counterflow state. During counterflow the system is still described by the $|Z\rangle$ eigenstate, and so if we now return ${B}_{z}$ to its initial value (or any other value of suitably large magnitude) the entire condensate will return to $|0\rangle$. We use this method for recombination in our interferometry protocol.

With our beam-splitting and recombination protocols established, we can now consider the impact of a relative phase-shift, consistent with the approach described in Halkyard et al. (2010). Artificially imprinting a relative phase difference $\delta$ between the spin up and spin down components after the split (at some time when ${B}_{z}=0$), we can re-write our counterflow state as a combination of all three eigenstates of ${\bf B}\cdot{\bf F}$.

[TABLE]

Ramping ${B}_{z}$ back up effects the recombination, after which the $|\pm B\rangle$ eigenvectors are the $|\pm 1\rangle$ states, while the $|Z\rangle$ eigenvector is the $|0\rangle$ state. Hence, projecting our final state onto the zero spin state via the $|0\rangle\langle 0|$ projector we obtain an interferometric signal based on the condensate fraction in the $|0\rangle$ state, i.e., $\int|\Psi_{0}(t_{\mathrm{f}})|^{2}\mathop{{\mathrm{d}}\boldsymbol{\mathrm{r}}}=[1+\mathrm{cos}(\delta)]/2$, where $t_{\mathrm{f}}$ is the time when our interferometry protocol ends. The populations of the different spin components can be observed experimentally by applying a field gradient in the $z$ direction, resulting in Stern–Gerlach separation.

We now consider prospective interferometry applications, where $\Omega_{z}$ or ${B}_{z}$ are non-zero during the interrogation counterflow period $T_{\mathrm{I}}$. We assume the counterflow state is well-described by Eq. (5), discarding structure and dynamics in the $\rho$ and $z$ directions; this assumption is validated by numerical simulation. If we apply the $i\hbar(\boldsymbol{\mathrm{r}}\times{\bf\Omega})\cdot\nabla$ operator to each spin component of the counterflow state [Eq. (5)], this yields eigenvalues $\pm\hbar\Omega_{z}$ in the $|\pm 1\rangle$ components respectively, and [math] in the $|0\rangle$ component. These eigenvalues can be incorporated into the diagonal elements of Eq. (2), combining the rotational and Zeeman terms of Eq. (1). The effect of the rotation is simply to offset the strength of the $z$ bias field, which is suppressed (enhanced) as the coordinate system rotates with (against) the magnetic dipole precession. This gauge transformation can be expressed as ${\tilde{B}}_{z}\rightarrow{\tilde{B}}_{z}+\tilde{\Omega}_{z}$, where ${\bf\tilde{B}}=\mu_{\mathrm{B}}g_{F}{\bf B}/\hbar\omega_{\perp}$ and ${\bf\tilde{\Omega}}={\bf\Omega}/\omega_{\perp}$ are dimensionless quantities. Experimentally, ${\tilde{B}}_{z}\gg{\tilde{B}}_{\mathrm{q}}\gg\tilde{\Omega}_{z}$ is typically achievable (and implicit in considering the rotation to be a “small effect”).

Such a transformed system has analogous transformed eigenvectors. Hence, ramping down $|{B}_{z}|\rightarrow 0$,we are still in the $|Z\rangle$ eigenstate of $\mathbf{B}\cdot\mathbf{F}$, and therefore expect no accumulation of phase difference between the $|\pm 1\rangle$ components. In order to observe relative phase accumulation, we must have a superpositional counterflow state in the absence of a quadrupolar field. Carefully ramping down ${B}_{\mathrm{q}}$ with ${B}_{z}=0$ achieves this aim and unpins the phases of the counterflowing components, but if some small residual contribution $B_{z{\mathrm{R}}}\neq 0$ remains then the system may return to $|0\rangle$ over a slow ${B}_{\mathrm{q}}$ ramp down as dictated by Eq. (4), and counterflow is lost. To avoid this restorative effect we must choose the ramp-down curve such that the ${B}_{\mathrm{q}}$ switch-off is diabatic in some sense. For example, it could be smoothly decaying at first and then cut off instantaneously before the point where ${B}_{\mathrm{q}}\eqsim 10\times B_{z{\mathrm{R}}}$, or be fully continuous but ramped over a suitably fast timescale. The key consideration should be reduction of the radial dynamics and heating associated with diabatic processes, noting that the smaller the residual field, the smaller the associated Zeeman energy, and so the less danger of heating. We also highlight that this restorative effect requires the residual field to satisfy $B_{z{\mathrm{R}}}\ll{B}_{\mathrm{q}}(R_{0})$ ($\sim 1.57$ mG for our weak $B$-field numerics). This upper bound scales linearly with $R_{0}$ and $b^{\prime}$, and can easily be raised. In the complete absence of magnetic fields (and, by spin-gauge symmetry, rotations) the hyperfine states become degenerate. In general this leads to undesirable spin-flips, which become more energetically allowable as $B\rightarrow 0$. Note that spin-flipping collisions are suppressed in the superpositional counterflow state Note (1). Another possible source of undesirable spin-flips is stray fields. However, assuming the field can be controlled on the mG scale, such processes have long associated timescales and can be ignored. Finally, we note that quantum and thermal fluctuations may be another source of spontaneous spin-flips, but such analysis is beyond the scope of this Letter. A strategy to avoid these spin-flips would be to purposefully retain a nonzero $B_{z{\mathrm{R}}}$.

Once the quadrupolar field is absent, the $|\pm 1\rangle$ components can evolve freely, and accumulate a Sagnac phase $\delta_{S}$ for $\Omega_{z}\neq 0$, or a Zeeman-energy phase $\delta_{\mathrm{Z}}$ for $B_{z{\mathrm{R}}}\neq 0$. The phase magnitude can be quantified in terms of either the ring’s enclosed area or the interrogation period $T_{\mathrm{I}}$ Halkyard et al. (2010). Allowing each component to perform the equivalent of one full circulation around the ring produces a Sagnac phase $\delta_{S}=4A\Omega_{z}m/\hbar$ Halkyard et al. (2010); Helm et al. (2015). The particle velocity around the ring is given by a vortex velocity-field, ${\bf v}=(\hbar/m\rho)\hat{\bf\phi}$, and so the time for a single particle to fully circumnavigate the ring (such that $\rho=R_{0}$) to be $T_{\mathrm{C}}=2\pi R_{0}^{2}m/\hbar=2Am/\hbar$ ($=313$ ms for our parameters). The phase accumulated for an arbitrary interrogation time $T_{\mathrm{I}}$ is then $\delta_{S}=(T_{\mathrm{I}}/T_{\mathrm{C}})4\Omega_{z}Am/\hbar=2\Omega_{z}T_{\mathrm{I}}$. The same arguments apply for the Zeeman-energy phase under the substitution $\Omega_{z}\rightarrow(\mu_{\mathrm{B}}/\hbar){B}_{z}$. After interrogation, restoring the quadrupolar $B$-field projects our phase-shifted wave-function onto the eigenstates of $\mathbf{B}\cdot\mathbf{F}$. As phases are accumulated the populations in the $|\pm B\rangle$, $|Z\rangle$ basis differ upon restoration of the quadrupolar field [Eq. (5)]. This induces some radial oscillations as the $|\pm B\rangle$ eigenstates are respectively strong and weak-field-seeking. These oscillations can be seen in the overlap integrals shown in Fig. 1(d), however they do not affect the recombination as the radial dynamics analytically decouple from the eigenvectors of $\mathbf{B}\cdot\mathbf{F}$.

We show results of numerical simulations of Eq. (1) (using CUDA Nickolls et al. (2008)) in Figs. 1 and 2. In  Fig. 1 we performed an interferometry procedure with Sagnac phase $\delta_{\mathrm{S}}=\pi$, fixing the interrogation times $T_{\mathrm{I}}=T_{\mathrm{C}}$ for complete circulation around the ring, and employing weak $B$-fields consistent with Ray et al. (2014). These fields require longer timescales, allowing us to better see the dynamics. Subplots (b), (c) and (d) display the full time evolution of the $B$-field, the norms $n_{i}=\int|\Psi_{i}|^{2}\mathop{{\mathrm{d}}\boldsymbol{\mathrm{r}}}$ of each component, and the density-density overlap integrals $\chi_{i,j}(t)=\int|\Psi_{i}|^{2}|\Psi_{j}|^{2}\mathop{{\mathrm{d}}\boldsymbol{\mathrm{r}}}$ respectively. There is a small difference between $n_{\pm 1}$ after recombination, as the weak fields used in these numerics make the Zeeman and nonlinear timescales comparable, compromising the dynamics. The result is still commensurate with our analytical predictions even in this sub-optimal regime. We observe good overlap during the counterflow phase, verifying that radial dynamics do not affect interrogation and that our method is a good example of a common path interferometer. After restoring the quadrupolar field oscillations are evident in the $\chi_{-1,+1}$ overlap integral as a result of the condensate now populating the field-sensitive $|\pm{B}\rangle$ eigenstates. In Fig. 2(a) we show the response curve obtained by varying $\Omega_{z}$ while again holding constant the interrogation time $T_{\mathrm{I}}=T_{\mathrm{C}}$. For these we used stronger $B$-fields ($b^{\prime}=37$ Gcm*-1* and ${B}_{z}=500$ mG). The response curve is smooth and in good quantitative agreement with our prediction [Eq. (5)]. We report that response curves obtained by varying $B_{z{\mathrm{R}}}$, for field sensing on the sub-$\mu$G scale, are in good quantitative agreement with those obtained by varying $\Omega_{z}$. In Fig. 2(b–d) we show calculations of the classical Fisher information $F_{\mathrm{C}}=\sum_{i=-1}^{1}(\partial_{\Omega}n_{i})^{2}/n_{i}$ and show that for $\Omega_{z}\neq 0$ it is approximately equal to the quantum Fisher information $F_{\mathrm{Q}}=4l^{2}T_{\mathrm{I}}^{2}$, the upper limit achievable for uncorrelated particles and so the upper limit available to mean-field treatments (Haine, 2016). The counterflow quantization is $l=F=1$ in our method. This confirms that our SOCI protocol maximises $F_{\mathrm{C}}$ for an arbitrary pre-selected read-off time.

For $\Omega\sim 0$ [Fig. 2 (d,b)], the value of $F_{\mathrm{C}}$ is dominated by the small number count in the $|\pm 1\rangle$ modes such that even small deviations from zero are highly undesirable, as is the case with all 2-mode interferometers. Our protocol can be designed to avoid this issue through the addition of an extra set of quadrupole bars to the IP coils. Using the secondary bars during recombination allows an arbitrary rotation of the quadrupole field about the $z$-axis, effecting a coordinate transformation equivalent to a phase shift $\delta$ [Eq. (5)]. Such a phase shift could move the response-curve to a more favourable location with $F_{\mathrm{C}}=F_{\mathrm{Q}}$. In this way, any rotation could be measured with precision limited only by the lifetime of the condensate. The sensitivity can be further increased by using a higher $F$ manifold, increasing $l$ and increasing precision. Note that the equivalency between $\tilde{\Omega}_{z}$ and ${\tilde{B}}_{z}$ requires that care be taken in experimental measurements. The maximum value of $\Omega_{z}=2\pi\times 5.0$ Hz used in the numerics corresponds to ${B}_{z}=3.56$ $\mu$G. As such, it should be straightforward to make single-shot field measurements on the sub-$\mu$G scale, as large rotations should be absent. Similarly, a spin-echo technique Hahn (1950) would allow the exclusion of Zeeman phases Halkyard et al. (2010).

In conclusion, we present a BEC interferometry protocol which requires only the careful control of standard $B$-fields and an optical ring-trap. Our protocol gives the greatest possible degree of access to measurement information for uncorrelated systems and, through its maximal spatial overlap, is a good candidate for Heisenberg limited interferometry Gross (2012); Ferris and Davis (2010); Ragole and Taylor (2016). We have also presented the results of full 3D multicomponent mean-field calculations of the Fisher information which demonstrate the robustness of our approach in the absence of idealizing approximations.

The data presented in this Letter can be found in Ref. dat .

We thank A. L. Marchant, R. J. Bettles, C. Weiss, and S. A. Haine for useful discussions, the UK EPSRC (grant number EP/K03250X/1) and the Leverhulme Trust (grant number RP2013-K-009).