A matter wave Rarity-Tapster interferometer to demonstrate non-locality

The system we use in our experiment is an atomic BEC of helium atoms in the long lived $2^{3}S_{1}$ metastable state. We initially form our BEC in a biplanar quadrupole Ioffe magnetic trap [50], which is well approximated as harmonic near its minimum. For our experiment we relax our trap such that it has trapping frequencies $\{\omega_{x},\omega_{y},\omega_{z}\}/2\pi\sim\{50,200,200\}$ Hz. Approximately $850$ mm below the trap we employ a micro-channel plate (MCP) and delay line detector (DLD) system which allows us to detect the positions of single atom impacts after the atoms are released from the trap and allowed to fall in free space under gravity [38]. Due to the distance the atoms travel, we can approximate these detection events as being in the far-field, enabling us to map the detected position of the atoms back to their initial velocity before expansion, with full 3D resolution. Thus, our system acts as a quantum many-body momentum microscope [48], allowing us to sample the momentum distribution of the in-trap state and making this an ideal system to measure momentum correlations.

A schematic of our experimental producedure to implement a matter wave Rarity-Tapster interferometer [10] is shown in Figs. 1 and 2. Our proposed setup is based on the original optical analog of the interferometer [Fig. 1(a)], which enables the measurement of phase-sensitive correlations between momentum-entangled twin-photons generated by a common source. This is achieved by mixing photons (or atoms) from independent pairs, labelled $(\mathbf{p},\mathbf{p}^{\prime})$ and $(\mathbf{q},\mathbf{q}^{\prime})$ in Fig. 1(a) (where we have used bold font to indicate vector quantities), at a pair of spatially separated beam-splitters after application of a pair of tunable phase-shifts in the upper arms of the interferometer. The trajectories through the interferometer to the detection ports $D_{\textbf{k}}$ ($\textbf{k}\in\{\mathbf{p},\mathbf{p}^{\prime},\mathbf{q},\mathbf{q}^{\prime}\}$) are such that it is not possible to distinguish the original pairs, which enables detection of interference between the possible paths that is sensitive to the applied phase shifts. In the optical version the applied phases ($\phi_{L}$ and $\phi_{R}$) is varied via optical elements inserted into the upper arms of the interferometer, in our case the phases will be operationally applied via the beamsplitter equivalent.

Our atomic analogue of this scheme is realized by applying a series of Raman and Bragg transitions, which utilise the $2^{3}P_{0}$ state, after switching off the confining trap. We first use a stimulated Raman transition, consisting of two collinear lasers of differing frequency, to transfer the BEC from the magnetically sensitive $2^{3}S_{1}\,\,\,m_{J}=+1$ state to the insensitive $2^{3}S_{1}\,\,\,m_{J}=0$ state [Fig. 2(i) inset] [18]. As the transition uses collinear lasers no momentum is imparted [18]. While the two transitions being stimulated are optimal for different polarisations we choose a single orientation that drives both transitions, i.e. is an equal combination of $\pi$ and $\sigma^{-}$ light, however, there is hence some power in the beam which is not used. To form our Bragg lattice, which allows us to control the momentum-state of the atoms (see Appendix D), we use two $276767$ GHz approximately collimated Gaussian laser beams (whose wavevectors we label $\textbf{k}_{u}$ and $\textbf{k}_{l}$) with $1/e^{2}$ radius of $0.7$ mm intersecting at an angle of $90^{\circ}$, with the resulting Bragg vector aligned along the $z$-axis (i.e., antiparallel to gravity). The spatial cross-section of the Bragg lasers is much larger than the atom paths ($0.7$ mm compared to a maximum atomic separation of $\sim 60\,\mu$m) hence providing a uniform illumination across the entire halos and atom path. As these beams intersect at right angles, they generate a two-photon momentum recoil of $2\hbar\textbf{k}_{0}=\hbar(\textbf{k}_{l}-\textbf{k}_{u})=\sqrt{2}\hbar k\hat{\textbf{z}}$, where $k$ is the wavenumber of the incoming beams ($|\textbf{k}_{u}|=|\textbf{k}_{l}|=k$), and $\hat{\textbf{z}}$ is a unit vector along the $z$-axis. For our chosen frequency of Bragg lasers, we thus have a Bragg vector of $2\textbf{k}_{0}=8.204\,\hat{\textbf{z}}$ $\mu$m^-1. We then apply, at $t_{0}$, a Bragg pulse that coherently splits the BEC into three momentum components, whose momenta have been respectively displaced by $+2\textbf{k}_{0},\,0,\,-2\textbf{k}_{0}$ [see Fig. 2(ii)]. We tune the pulse such that the populations of the three momentum modes are approximately equal.

As the moving condensates spatially separate, collisions between constituent atoms will occur. Provided that the BECs are in the slow moving regime these will be limited to $s$-wave, spherically symmetric elastic scattering events. From these, a pair of so-called $s$-wave collision halos form in momentum space that, as a result of conservation of momentum and energy, are nearly spherical shells centered on the center-of-mass (COM) of each of the adjacent BEC pairs [Fig. 2(iii) and 1(b)] [51, 48]. More precisely, the collision halos are composed of an ensemble of independent two-mode-squeezed states, with each pair of squeezed modes occupying diametrically opposed momenta due to conservation of momentum and energy, which can be used as the input state for our matter wave Rarity-Tapster interferometer. Collisions between the $+2\textbf{k}_{0}$ and $-2\textbf{k}_{0}$ condensates also generate a larger halo, but this does not participate in the remainder of the dynamics and for clarity we omit it from Fig. 2. We point out that our proposal to leverage twin-atoms generated in two collisonal halos is an important distinction from the previous theoretical proposal (based on a single halo) by two of the authors, Ref. [44], and we elaborate on the consequences in Sec. II.3.

After the halos are populated (we assume they are populated up to time $t_{1}$) we apply a series of Bragg pulses to couple a select set of momentum modes such that their paths through the interferometer are indistinguishable, and hence interfere. We refer to this setup as a Rarity-Tapster type matter wave interferometer [10] as it operates on the same principle as the optical version. Figure 2(iii-v) shows the momentum space picture and Fig. 1 shows the trajectories through position space for this interferometer, with comparison to the optical counterpart. The timings of the pulses (mirror pulse at $t_{2}$ and beamsplitter pulse at $t_{3}$), equivalent to the spatial positioning of the mirrors in the optical interferometer, must be optimised so that they provide maximum overlap between the scattering pairs, both in momentum and position space [44].

As previously mentioned, the collision halos are ideally composed of an ensemble of entangled twin-atoms with opposing momenta. Here, we elaborate on this statement by first considering the upper collision halo (formed by atoms scattering from the BECs with momenta $0$ and $+2\mathbf{k}_{0}$), before discussing how twin-atoms, scattered into the distinct upper and lower collision halos, approximately realize a prototypical Bell state that can be used to demonstrate violations of a Bell inequality. Specifically, the mode-entanglement between squeezed modes will be turned into particle-like entanglement between a quartet (two pairs) of modes via post-selection, which is practically approximated using the low-gain regime.

The collision of a pair of condensates realizes a matter-wave analog of four-wave mixing from quantum optics [52, 53]. Moreover, if the finite momentum width of the prepared condensates and depletion of its occupation due to the scattering of atoms into new momentum states is ignored, then we can approximately describe the resonant scattering of atoms from a pair of colliding condensates using a sum of two-mode squeezing Hamiltonians, $\hat{H}=\sum_{\textbf{p}}\hbar\zeta\left(\hat{a}_{\textbf{p}}^{\dagger}\hat{a}_{\textbf{p}^{\prime}}^{\dagger}+\hat{a}_{\textbf{p}^{\prime}}\hat{a}_{\textbf{p}}\right)$ with $\textbf{p}^{\prime}=-\textbf{p}+2\textbf{k}_{0}$. Here, $\hat{a}_{\mathbf{p}(\mathbf{p}^{\prime})}^{\dagger}$ creates a boson with momentum $\mathbf{p}\,(\mathbf{p}^{\prime})$ and $\zeta$ is an effective nonlinearity that is dependent on the $s$-wave scattering strength and the colliding BECs’ densities [44], assumed to be uniform for simplicity. Note that the momenta of the scattering pair must satisfy the condition $\textbf{p}+\textbf{p}^{\prime}=2\textbf{k}_{0}$, where $2\textbf{k}_{0}$ is the sum of the colliding BECs’ momenta, as we have assumed resonant scattering (e.g., energy is precisely conserved) and ignored the momentum width of the source BECs. Note, further that if we consider the system with respect to the COM frame of the halo, then our description of the scattered pairs reduces to that of previous works [44, 26].

Considering now the collision of $0$ and $+2\mathbf{k}_{0}$ BECs and assuming an initial vacuum state for all momenta $(\mathbf{p},\mathbf{p}^{\prime})\neq 0$ or $2\mathbf{k}_{0}$, the two-mode squeezing Hamiltonian generates a product of two-mode squeezed vacuum states for the upper halo, which can be expressed in the Fock basis as [54, 55],

where $\mathbf{p}^{\prime}=-\mathbf{p}+2\mathbf{k}_{0}$, $\mu=\tanh(\zeta t_{1})$ for a collision duration $t_{1}$, and $\zeta t_{1}$ is the effective squeezing parameter. Thus, Eq. (1) represents a product state composed of superpositions of number-correlated states $\ket{n}_{\mathbf{p}}\ket{n}_{\mathbf{p}^{\prime}}$ (in any pair of modes with momenta $\mathbf{p}$ and $\mathbf{p}^{\prime}$ satisfying $\mathbf{p}+\mathbf{p}^{\prime}=2\mathbf{k}_{0}$) with an increasing $n$ and a decreasing probability amplitude $\propto\!\mu^{n}$, where $\mu<1$. Further, if we consider a single mode, its occupation is described by thermal counting statistics [56]. The mean number of atoms in a particular scattering mode is given by $\bar{N}=\frac{\mu^{2}}{1-\mu^{2}}=\sinh^{2}(\zeta t_{1})$. Therefore, $\bar{N}\approx\mu^{2}\ll 1$ for $\mu\ll 1$, i.e., in the low-gain or perturbative regime that we are going to consider below.

Under the same assumption as before—that the depletion of colliding condensates is negligible and the initial state of the halo is a vacuum state—the state of the second (lower) halo, which is formed by the collision between the $-2\textbf{k}_{0}$ and $0$ BECs, is independent of the upper halo. Therefore, the lower halo can be identically described by Eq. (1), albeit with corresponding $\mu^{\prime}=\tanh(\zeta^{\prime}t_{1})$,

where $\textbf{q}^{\prime}=-\textbf{q}-2\textbf{k}_{0}$. We introduce the distinct $\mu^{\prime}$ to incorporate the possibility that, e.g., the density of the split BECs is not identical (leading to a different nonlinearity $\zeta^{\prime}$) but assume the collision duration $t_{1}$ is unchanged. If all three colliding BECs are identical then $\mu=\mu^{\prime}$, which we will assume herein. The independence of the formed collision halos implies that the complete state describing all the (resonantly) scattered atom pairs in both halos is

Now we will show how the input state of our interferometer, a reduced form of $\ket{\Psi}_{\text{double-halo}}$, can be approximated to one of the prototypical Bell states. To do so, we focus on just two correlated pairs of momentum modes $(\mathbf{p},\mathbf{p}^{\prime})$ and $(\mathbf{q},\mathbf{q}^{\prime})$ selected from the upper and lower halos, respectively, such that they satisfy,

and simultaneously are related by,

We illustrate an example of the selected modes in Figs. 1 and 2. Physically, the modes correspond to two scattering pairs $(\textbf{p},\textbf{p}^{\prime})$ and $(\textbf{q},\textbf{q}^{\prime})$, which are exactly separated by the two-photon momentum recoil $2\textbf{k}_{0}$ used in the initial momentum slitting of the source BEC [see Fig. 2(ii)]. They were chosen as they are the only modes, within the halos, which exactly match the Bragg condition, i.e., are separated by an integer number of lattice wave vectors, for the geometry used to generate the original BEC momentum splitting. This allows for a more stable and simple implementation, as discussed in further detail below.

Ignoring or tracing away all other momentum mode pairs from $|\Psi\rangle_{\mathrm{double-halo}}$ beyond the two pairs considered, the factorised state of the four modes of interest can be written as

Assuming now that the scattering from the condensates is in the low-gain, perturbative regime, $\mu\ll 1$, we can truncate the sum over Fock states to lowest order in $\mu$, equivalent to ignoring the contribution of Fock states with the total occupancy among the four modes larger than 2 (i.e., restricting ourselves, via post-selection, to no more than a total of 2 particles across the two halos) The removal of higher occupancy terms has the effect of entangling the pair of selected squeezed modes $(\mathbf{p},\mathbf{p}^{\prime})$ with the other selected pair $(\mathbf{q},\mathbf{q}^{\prime})$ , which previously had been independent of each other (non-entangled). It will be seen that this is why in the low-gain limit ($\mu\rightarrow 0$) the full state, despite being factorisable, is able to approximate a maximally particle entangled Bell state. We enforce the low-gain, perturbative regime used in this analysis, and the experiment, by ensuring that $\zeta t_{1}\ll 1$, which typically corresponds to either a short effective collision duration (due to, e.g., rapid spatial separation of the BECs) or low initial density of the source condensate. The resulting truncated state for the two pairs of correlated modes can therefore be written as

As the vacuum state $\ket{0}_{\mathbf{p}}\ket{0}_{\mathbf{p^{\prime}}}\ket{0}_{\mathbf{q}}\ket{0}_{\mathbf{q}^{\prime}}$, the first term in Eq. (7), does not contribute to any measurement of population correlations that we will consider below, we can further truncate the above expression to

where we have enforced normalization.

This reduced and truncated state, describing the atom pairs in modes $(\mathbf{p},\mathbf{p}^{\prime})$ and $(\mathbf{q},\mathbf{q}^{\prime})$ in the low-gain regime, is formally equivalent to a paradigmatic polarization-entangled Bell state of two photons exploited in optical tests of quantum nonlocality. To make this clear we consider a mapping of atomic mode occupations in the left (L) and right (R) arms of the Rarity-Tapster interferometer [see Figs. 1 (a-b) and 2 (iii)] onto two, horizontal (H) and vertical (V), polarisation states of photons,

corresponding to an atom in the left arm populating either the $\mathbf{q}$ or $\mathbf{p}$ momentum mode. Similarly, for the right arm,

corresponding to an atom in the right arm populating either the $\mathbf{q}^{\prime}$ or $\mathbf{p}^{\prime}$ momentum mode. Then, Eq. (8) can be mapped to $\ket{\Psi}\rightarrow|\Psi\rangle_{\mathrm{Bell}}=\frac{1}{\sqrt{2}}(|H\rangle^{(\mathrm{L})}|H\rangle^{(\mathrm{R})}+|V\rangle^{(\mathrm{L})}|V\rangle^{(\mathrm{R})})$, or

This can now be readily identified as the prototypical Bell state, which is known to maximally violate a Bell inequality [9, 57, 58]. While for our purposes we will only consider pairs selected from different halos, any two independent pairs of correlated modes, across either halo, can be mapped to one of the Bell states after post-selection. Which Bell state exactly depends on the interferometeric geometry used; for example, the state under the previously proposed scheme of Ref. [44] will map to $\frac{1}{\sqrt{2}}(\ket{H,V}+\ket{V,H})$.

We again wish to highlight that the entanglement in our reduced state $\ket{\Psi}$, Eq. (6), is identifiable as mode-entanglement, which is between back-to-back (diametrically opposed) momentum modes in a single halo. Only after we remove the contribution of states with single mode occupation higher than or equal to $2$, which is equivalent in a physical sense to post-selecting for states that only contain a single pair of atoms among all four considered modes, can the state be reinterpreted as featuring particle Bell-state entanglement. In practice, robust post-selection is not possible due to the low detection efficiency that can be achieved in our experimental setup. Thus, our atomic four-wave mixing source acts as a probabilistic generator of particle entangled Bell states, rather than a deterministic one as is usually considered. Hence, the need for many experimental runs in the low mode occupation regime; we require that in the relevant momentum modes the probability of mode occupancies of $2$ or higher is negligible, while retaining a small probability of desired occupancies of $1$ in a fraction of experimental runs.

The working principle of the matter wave Rarity-Tapster interferometer in our realization is as follows. The pairs of entangled atoms with momenta $(\mathbf{p},\mathbf{p}^{\prime})$ and $(\mathbf{q},\mathbf{q}^{\prime})$ are reflected onto spatially separated atomic beamsplitters with phases $\phi_{L}$ and $\phi_{R}$. The atomic beamsplitters cause the initially distinct entangled pairs to become indistinguishable and leads to multi-particle interference effects that are sensitive to the applied phases (see Fig. 1). We note that to exploit this multi-particle interference for a proper demonstration of quantum nonlocality we in principle require independent tunability of the phases $\phi_{L}$ and $\phi_{R}$ (see Sec. II.5). However, in our current realization we realize the atomic beamsplitters with a single Bragg laser, such that $\phi_{L}=\phi_{R}$, which is sufficient for us to benchmark the interferometer. Our matter wave interferometer is equivalent in spirit to that employed by Rarity and Tapster in their experimental violation of the Bell inequality using the phase and momentum of photons [59, 10], but with a swapping of particle labels in one arm of the interferometer.

At the output of the interferometer (i.e., at some time $t_{4}$ after the final beam-splitter) we measure population correlations between the two pairs of momenta $(\mathbf{p},\mathbf{p}^{\prime})$ and $(\mathbf{q},\mathbf{q}^{\prime})$. To be concrete, we introduce the pair correlation function $C_{\mathbf{k},\mathbf{k}^{\prime}}=\langle\hat{a}^{\dagger}_{\mathbf{k}}\hat{a}^{\dagger}_{\mathbf{k}^{\prime}}\hat{a}_{\mathbf{k}^{\prime}}\hat{a}_{\mathbf{k}}\rangle$ (where $\textbf{k}\in\{\textbf{p},\textbf{q}\}$ and $\textbf{k}^{\prime}\in\{\textbf{p}^{\prime},\textbf{q}^{\prime}\}$), which corresponds to a correlation between joint-detection events at the outputs of the left and right arms of the matter wave interferometer (see Figs. 1 and 2). In simple terms the two point correlation function gives us a measure of the probability of finding a similar number of particles at momenta k and $\textbf{k}^{\prime}$. It hence contains information about the spatially separate interference and will serve as the basis of our test of Bell’s inequality.

Treating the atomic mirror and beamsplitter elements of the interferometer as instantaneous linear transformations (see Appendix A for details of the calculation), and using the reduced state, Eq. (6), as input, we obtain the normalized pair-correlations

where $\bar{N}=\frac{\mu^{2}}{1-\mu^{2}}$ is the average occupancy of any of the four modes, $\mathbf{p}$, $\mathbf{p}^{\prime}$, $\mathbf{q}$, and $\mathbf{q}^{\prime}$. Here, we emphasize that interference between the scattered pairs, as captured by the sine and cosine terms in the above pair-correlation functions, is tuned via the global phase $\Phi=\phi_{L}+\phi_{R}$, rather than the relative value of $\phi_{L}-\phi_{R}$.

In the regime $\mu\ll 1$ (i.e., vanishing probability of multiply occupied modes, $\bar{N}\ll 1$) the phase-sensitive terms $\propto\mu^{-2}$ dominate the correlation functions. However, when the mode occupancies with 2 or more atoms become appreciable, corresponding to $\bar{N}>1$ ($\mu\lesssim 1$), we find that the oscillatory terms depending on the global phase $\phi_{L}+\phi_{R}$ diminishes relative to the background and thus interference is suppressed. Nevertheless, the undepleted pump approximation, used to derive Eq. (6) and which ensures a highly nonclassical state of the scattered pairs, will eventually fail in the high-gain, nonperturbative limit.

We will find that this demonstrates that the matter wave Rarity-Tapster interferometer optimally probes quantum nonlocality when the initial state of the scattering halo can be closely approximated to a prototypical Bell state, i.e., when $\mu\ll 1$.

It is interesting to note that, as we do not experimentally post-select on our final state, the modes $(\mathbf{p},\mathbf{p}^{\prime})$ by definition remain unentangled with $(\mathbf{q},\mathbf{q}^{\prime})$ as the state is factorisable. Specifically, there is no entanglement between states in different halos. Nonetheless, it can be seen from Eqs. 14 and 15 that the mode-entanglement between $\mathbf{p}$ and $\mathbf{p}^{\prime}$, as well as between $\mathbf{q}$ and $\mathbf{q}^{\prime}$, in the initial state (i.e. between the left and right arms of the interferometer) is sufficient to violate Bell’s inequality for sufficiently small $\mu$.

Experimentally, the finite momentum and spatial width of the initial BECs contributes to a finite correlation width of the scattered atoms, which may no longer be scattered with precisely correlated momenta as in Eq. (4). To account for this, we first introduce the integrated pair-correlation function between detection regions centered around $\textbf{L}\in\{\textbf{p},\textbf{q}\}$ and $\textbf{R}\in\{\textbf{p}^{\prime},\textbf{q}^{\prime}\}$ (see Figs. 1 and 2 for definition of port labeling), which is defined as,

where $G^{(2)}(\textbf{k},\textbf{k}^{\prime},t_{4})\!=\!\left\langle\hat{a}^{\dagger}(\textbf{k},t_{4})\hat{a}^{\dagger}(\textbf{k}^{\prime},t_{4})\hat{a}(\textbf{k}^{\prime},t_{4})\hat{a}(\textbf{k},t_{4})\right\rangle$ is the two-point momentum correlation function evaluated at time $t_{4}$ after the interferometric sequence, $V(\textbf{L})$ and $V(\textbf{R})$ are the volumes of the respective detection regions in momentum space, whereas $\hat{a}^{\dagger}(\textbf{k},t_{4})$ and $\hat{a}(\textbf{k},t_{4})$ are the Fourier components of the field creation and annihilation operators describing the scattered atoms, evaluated at time $t_{4}$ in the Heisenberg picture [44]. Note that $\mathcal{C}_{\textbf{L},\textbf{R}}$ can be considered a generalization of $C_{\mathbf{k},\mathbf{k}^{\prime}}$ where momenta are no longer treated as discrete.

The evaluation of $\mathcal{C}_{\textbf{L},\textbf{R}}$ requires us to obtain a specific form for $G^{(2)}(\textbf{k},\textbf{k}^{\prime},t_{4})$ for cases where $\textbf{k}\approx\textbf{k}^{\prime}\pm 2\textbf{k}_{0}$, and the sign depends on the relevant pair of detection regions under consideration. To do so, we again treat the Bragg pulses as instantaneous (and perfect) linear transformations. Moreover, we adopt a Gaussian approximation for the two-point correlation $G^{(2)}(\textbf{k},\textbf{k}^{\prime},t_{1})$ of the input state [44, 60, 61] between pairs of momenta $(\mathbf{k},\mathbf{k}^{\prime})$ within the same scattering halo,

Here, $h$ is the height of the correlation amplitude above the background level (generally assumed to be $1$), $\sigma_{d}$ is the momentum correlation width in dimension $d$ ($d=x,y,z$) of the original state, $\delta_{ij}$ is the Kronecker delta function, and $n_{0}$ is the peak momentum space density of scattered atoms. Note, if we take the two-mode squeezed state to characterize the halo, the correlation height $h$ is related to the mode occupancy as $h=1/\mu^{2}\approx 1/\bar{N}$, and $n_{0}\prod_{d}\sigma_{d}=\bar{N}$. The plus and minus signs in Eq. (17) correspond to correlations between modes in the upper and lower halos, respectively. For pairs of momenta $(\mathbf{k},\mathbf{k}^{\prime})$ selected from distinct halos we have that $G^{(2)}(\textbf{k},\textbf{k}^{\prime},t_{1})/n_{0}^{2}=1$, as the scattered halos are assumed to be independent and uncorrelated.

Putting these pieces together, we obtain the relevant two-point correlation functions at the conclusion of the matter wave interferometer,

where the sign of the cosine function depends on the particular detection port combination. Again, the sign of the term in the Gaussian depends on whether the relevant momenta are in the upper ($-$) or lower ($+$) halos. In addition to the Gaussian dependence on the correlation lengths that is inherited from $G^{(2)}(\textbf{k},\textbf{k}^{\prime},t_{1})$, a momentum-dependent phase offset $\varphi(\textbf{k},\textbf{k}^{\prime})$ (see Appendix B) between off-resonant scattered pairs is included. We postpone a discussion of the physical meaning of this term to below.

The relevant integrated pair-correlation function, Eq. (16), is evaluated by substituting Eq. (18) and integrating over a detection bin with dimensions $\Delta k_{d}$ for $d=x,y,z$, leading to the result

We have introduced the average number of particles in the integration volume, $\bar{N}_{V}=n_{0}\prod_{d}\Delta k_{d}$, and $\lambda_{d}\equiv\Delta k_{d}/(2\sigma_{d})$ as the normalized resolution of the detection bin (see Tab. 1 for experimental values). Compared to the simpler form of Eq. (14), the function $\alpha_{d}$ for $d=x,y$ takes into account the contribution of the finite correlation widths in Eq. (18), while the function $\beta_{d}$ for $d=z$ also includes an additional contribution from the phase factor $\varphi(\textbf{k},\textbf{k}^{\prime})$ (which is in fact only a function of $k_{z}$ and $k_{z}^{\prime}$ for our interferometer configuration). These functions are defined as follows,

and,

where $A_{d}=\frac{k_{0}\sigma_{d}\hbar}{m}(\frac{t_{3}}{2}-t_{1})$ and $D_{F}(x)=e^{-x^{2}}\int_{0}^{x}e^{y^{2}}dy$ is the Dawson $F$ function.

The dependence of $\mathcal{C}_{\textbf{L},\textbf{R}}$ on the size of the chosen detection bins captures the fact that for $\lambda_{d}\gg 1$ there are many uncorrelated atoms within each integration volume, and so we expect $\mathcal{C}_{\textbf{L},\textbf{R}}/\bar{N}_{V}^{2}\to 1$. However, the integrated pair-correlation also encodes a subtle dependence on the relative timing of the mirror and beam-splitter Bragg pulses via the $\beta_{z}$ factor, and in turn the momentum dependent phase offset $\varphi(\textbf{k},\textbf{k}^{\prime})$ in Eq. (18), from which it arises. This phase offset captures the importance of indistinguishability in the atomic realization of the Rarity-Tapster interferometer, in close analogy to work studying Hong-Ou-Mandel interference with atom pairs [37]. In particular, we see that $A_{z}\to 0$, and consequently $\beta_{z}\to\alpha_{z}$, by choosing $t_{3}=2t_{2}$, i.e., by setting the free propagation period $t_{3}-t_{2}$ between the mirror pulse ($t_{2}$) and final beamsplitter pulse ($t_{3}$) to be identical to the prior time period between the commencement of the BEC collision and the application of the mirror pulse. This result is distinct to, and in fact much simpler than, that found in the prior study of Ref. [44], which considered correlated pairs of momentum modes drawn from a single scattering halo. In that case, the optimal free propagation period $t_{3}-t_{2}=t_{2}-m\sigma_{g,z}/k_{0}\sqrt{\pi}$ was offset by a factor equivalent to the effective collision duration set by the time taken for the colliding BECs to spatially separate, $t_{\mathrm{sep}}=\sigma_{g,z}m/\hbar k_{0}$ where $\sigma_{g,z}$ is the estimated rms spatial width of the initial unsplit condensate. This distinction between our scheme exploiting two scattering halos and that of Ref. [44] arises due to a subtle difference in the relative position along the splitting direction ($z$-axis) at which the scattered pairs are considered to be created as a function of collision duration.

In our scheme, pairs located on, e.g., the equatorial $k_{x}-k_{y}$ plane of the scattering halo are always created at the COM of the colliding BEC pair, with each scattered pair also having COM momenta $\pm k_{0}\hat{\textbf{z}}$. Thus, if we consider the creation of scattered pairs as a classical stochastic process over time, all the created pairs have an identical COM position along the $z$-axis and travel with the same velocity along the direction of the applied Bragg pulses. This means that for $t_{3}=2t_{2}$ all scattered pairs share approximately the same $z$ position (up to, e.g., corrections due to the initial spatial width of the source condensates) when they are subject to the final beamsplitter Bragg pulse. Importantly, this implies that it is not possible to distinguish which pair of the originally coupled momentum modes, $(\mathbf{p},\mathbf{p}^{\prime})$ and $(\mathbf{q},\mathbf{q}^{\prime})$, an atom belongs to when it is finally detected and the integrated pair-correlations $\mathcal{C}_{\textbf{L},\textbf{R}}$ are constructed. This indistinguishability between the possible paths in the interferometer is pivotal to obtain phase-sensitive correlations, which is immediately seen by noting that $\mathcal{C}_{\textbf{L},\textbf{R}}/\bar{N}_{V}^{2}\to 1$ for $A_{d}\gg 1$ (corresponding to a large mismatch between the free propagation periods before and after the mirror Bragg pulse).

In contrast, in Ref. [44] scattered pairs are always created approximately at the origin but at random times $0\leq t\lesssim t_{\mathrm{sep}}$, leading to an intrinsic spatial distribution of pairs along the direction of the applied Bragg pulses. Averaging over the time of creation of the pairs results in the optimal offset between free propagation times before and after the applied mirror Bragg pulse. While this distinction may appear to only be a minor detail, it is important to emphasize that by using a pair of scattering halos in this manner our scheme does not require exhaustive calibration of the free propagation time. In contrast, the offset $t_{\mathrm{sep}}$ in Ref. [44] is derived under an assumption of a Gaussian density profile for the colliding BECs and would require experimental optimization.

Ultimately, the phase-sensitive correlations encoded by $\mathcal{C}_{\textbf{L},\textbf{R}}$ can be used as part of a Bell inequality to demonstrate the inconsistency of some predictions of quantum mechanics with local hidden variable theories. First, we construct the quantum correlator,

which depends only on the global phase $\Phi=\phi_{L}+\phi_{R}$ imprinted during the interferometer. From this, one obtains the CHSH-Bell parameter $S$ for the Clauser-Horne-Shimony-Holt (CHSH) version of the Bell inequality [10, 62], where

which is bounded by $S\leq 2$ for any local hidden variable theory. For $E(\phi_{L},\phi_{R})$, as given by Eq. (23), we find that it is possible to violate this inequality for specific choices of phase settings [10, 44] and instead we are limited by $S\leq 2\sqrt{2}$. As our interferometer depends on the global phase $\phi_{L}+\phi_{R}$, as opposed to the phase difference as in Refs. [10, 44], we note that the optimal phase settings are different to the typical choice by a sign, $(\phi_{L},\phi^{\prime}_{L},\phi_{R},\phi^{\prime}_{R})=(0,\pi/2,-\pi/4,-3\pi/4)$ ¹¹1For complete clarity, we note that the difference in phase sensitivity in our scheme to, e.g., Refs. [10, 44], is due to an effective switching of which arm of the interferometer one of the phase shifts is imprinted.. The saturation of the quantum bound, $S=2\sqrt{2}$, occurs when we take $\lambda_{x,y,z}\to 0$ and $h\to\infty$, which corresponds to the limit where the twin-atoms of the scattering halo can be approximately mapped to the prototypical Bell state [Eq. (13)]. The conditions for this to occur typically correspond to a very low population of the scattering halo and large correlation length. Equation (23) is the main result of our analytic model, and the dependence on $\Phi$ and $\lambda_{x,y,z}$ is compared to experimental data in the following section (see Figs. 3 and 4, respectively).