We are now ready to analyze the behavior of the first beam splitter acting in the interferometric process. We perform our study for two different initial states: a Twin Fock state

$|N/2,N/2\rangle $ (

Section 4.1) and a

$|N,0\rangle $ state (

Section 4.2). For the sake of comparison, we also study the sensitivity reached when a NOON state,

$(|N,0\rangle +|0,N\rangle )/\sqrt{2}$, is created after the beam splitter (

Section 4.3).

The general process used to realize a beam splitter for an atomic interferometer is to let the particles tunnel after lowering the inter-well barrier in the case of a double-well interferometer, or by coupling the two states with resonant light, in the case of an internal two-level system interferometer. In characterizing the beam splitter, we use the sensitivity as the full width at half maximum (FWHM) of the narrowest peak in the phase probability distribution. The latter can be calculated as [

2]:

where

is the state after the first beam splitter. The coefficients

${c}_{n}(t)$ are given by

${c}_{n}(t)=\langle n|{\psi}_{out}(t)\rangle $,

$\left\{\right|n\rangle \}$ with

$n=[0,N]$ being the

$N+1$ Fock basis vectors for the two-mode model:

$|n\rangle ={|n\rangle}_{a}{|N-n\rangle}_{b}$. The

$|\varphi \rangle $ are the normalized phase states

$|\varphi \rangle ={(N+1)}^{-1}{\sum}_{m=0}^{N}{e}^{i\varphi (N/2+m)}|m\rangle $.

#### 4.1. Initial State Twin Fock

When the initial state is a Twin Fock (TF) state

$|N/2,N/2\rangle $, the optimal splitting time is defined as the time after which the main peak in the phase probability distribution is the narrowest [

2]. In the non-interacting case, the

$50/50$ beam splitter is represented by the unitary transformation

$|{\mathsf{\Psi}}_{out}\rangle ={e}^{-i\frac{\pi}{2}(\frac{{\widehat{a}}^{\u2020}\widehat{b}+{\widehat{b}}^{\u2020}\widehat{a}}{2})}|{\mathsf{\Psi}}_{in}\rangle $ and the optimal splitting time is given by

${T}_{BS}=\pi /(4J)$. The latter corresponds to a

$\pi /2$ Raman pulse, which in the following we will refer to as

${T}_{\pi /2}$. As discussed in [

2], the sensitivity is at the Heisenberg limit, i.e.,

$\Delta \varphi =\alpha /{N}^{\beta}$ with

$\beta \sim 1$: from our numerical simulation, we obtain

$\beta =1.001\pm 0.004$ and

$\alpha =5.03\pm 0.05$. When local interactions are present (

$U\ne 0$,

$V=0$), the phase probability distribution broadens and smaller values of

$\beta $ are found after the beam splitter, with respect to the non-interacting case: from our numerical simulation we fit the parameters values

$\alpha =1.1\pm 0.1$ and

$\beta =0.53\pm 0.02$. Finally, we observe that the optimal splitting time decreases as the interaction energy is increased [

2].

Let us now consider the effect of inter-well interactions. As already noticed, in the limit

$U/J\to -\infty $ the ground state of Hamiltonian (

1) is a NOON state. However, when the intra-well interactions are attractive, a collapse of the atomic cloud would take place, if the number of atoms exceeds a critical value [

69]. Varying the inter-well interactions far from the value

$V=U$, the optimal splitting time decreases while the width of the peak in the phase probability distribution increases, as illustrated in

Figure 3. The optimal splitting time decreases with increasing

N at fixed

U, and it is progressively less dependent on the number of particles while the intra-well interactions get larger. For

$V=U$, the narrowest peak in the phase probability distribution has the same width for each value of

U. As predicted at the beginning of this section, the scaling parameter

$\beta $, defined from

$\Delta \varphi =\alpha /{N}^{\beta}$, is equal to that for the non-interacting case, in particular

$\alpha =5.03\pm 0.05$ and

$\beta =1.001\pm 0.004$. In

Figure 4, the scaling parameter

$\beta $ for different

U is reported against

$V/U$: we notice that

$\beta $ decreases with increasing the interactions, except for

$V=U$, where the non-interacting case is recovered.

#### 4.2. Initial State $|N,0\rangle $

We now consider the splitting process for the initial state

$|N,0\rangle $, where the system is prepared with all the particles in one well. In this case, we define the optimal splitting time as the time at which the population is equally split among the two wells (

$z=0$). We first analyze the non-interacting case. As for the Twin-Fock case as input state, the optimal splitting time is given by

${T}_{BS}=\pi /(4J)$, once again referred to as

${T}_{\pi /2}$. For

$U=V=0$, the shot-noise sensitivity is reached and

$\Delta \varphi =\alpha /{N}^{\beta}$, with

$\beta \sim 0.5$. When considering

$U\ne 0$ (with

$V=0$), the condition

$UN/J\le 4$ must be fulfilled in order to avoid self trapping. Interactions are found to broaden the peak in the phase probability distribution, as shown in

Figure 5.

After increasing the particles number, the width of the phase probability distribution peak decreases when

N is far from the self-trapping limit. Approaching the self-trapping limit, instead, the width begins to rise until the self-trapping threshold is reached. As a result, a minimum is found at intermediate values of

N, as visible in the left panel of

Figure 6. This can be understood by looking at the phase probability distributions for different values of the particles number. By increasing

N near the self-trapping limit, the peak shifts and broadens. As displayed in

Figure 7, at the self-trapping threshold the main peak splits in several sub-peaks, resulting in a narrower width. A similar behavior is found when the self-trapping threshold is reached by increasing the value of the interaction energy at fixed particles number.

$\Delta \varphi $ increases for energies lower than the self-trapping value and a minimum is found at the self-trapping threshold. This is illustrated in the right panel of

Figure 6.

As to the optimal splitting time for the case of

$|N,0\rangle $ as input state, this is found to increase with increasing

$UN$, its values resulting in being the same for both repulsive and attractive interactions. This behavior is visible in

Figure 8, where the optimal splitting time is displayed while varying

U at fixed

N. Similar results are found by fixing

U and varying

N. Now, the red curve in

Figure 8 is given by the terms up to quadratic in the approximated equation for

${t}_{optimal}$:

obtained in terms of

$\mathsf{\Lambda}\equiv UN/(2J)$ after integrating the equation for the population imbalance evolution

between

$z(0)=1$ and

$z(t)=0$ and by scaling the time in units of

$\pi /(4J)$. The slight discrepancy between fit and theory can be attributed to the small value of the particles number considered here.

Finally, looking for a universal dependence on

$UN/(2J)$, the optimal splitting time has been found to be dependent on the number of particles at fixed interaction energy (or viceversa) for small

N (large

U), as shown in

Figure 9. This behavior holds only for small values of the particles number, as we can infer from the inset in the left panel of

Figure 9. By increasing the value of the particles number, the difference between the corresponding optimal times decreases. In the right panel of

Figure 9, the optimal splitting time is plotted for different values of the particles number: we see that when

N is small, the self-trapping threshold is higher than that prescribed by Equation (

7), valid for

$N\gg 1$.

We now turn to consider the effects of inter-well interactions. Varying the inter-well interactions, the self-trapping limit can be reached for intra-well energy lower than the one satisfying Equation (

7) (

$UN/(J)\le 4$), as shown in the lower panel of

Figure 10. Indeed, we can see the self- trapping effect in the curve for

$N=20$: for

$V/U=-0.5$ and

$V/U=2.5$. The curve presents maxima with values higher than the values found from the curves with

$N=18$ and

$N=16$. Notice that near the self-trapping limit, an increase in the particles number leads to increasing values of

$\Delta \varphi $, as displayed in

Figure 6. Local minima are found for the values

$V/U=-1$ and

$V/U=3$, corresponding to

$U-V=\pm 0.2$ in the Hamiltonian (

15), that is the self-trapping threshold.

We close this analysis by noticing that the narrowest peak in the phase probability distribution for

$V=U$ has the same width for each value of

U. Similarly, the scaling parameter

$\beta $ (

$\Delta \varphi =\alpha /{N}^{\beta}$) has the same value found in the non-interacting case, as reported in the right panel of

Figure 11. As it is evident from the left panel of

Figure 11 displaying

$\beta $ vs $V/U$ for different

U values, increasing interactions worsen the scaling behavior, except when

$V=U$, where the non-interacting case is recovered.

#### 4.3. Discussion

We conclude this section by discussing the comparison between the sensitivity that could be reached, if a NOON state

$|NOON\rangle =|N,0\rangle +|0,N\rangle /\sqrt{2}$ be created after the beam splitter. Though creating a NOON state is a challenging task [

1], this is a useful analysis, since NOON states are known to provide a very good sensitivity in atom interferometry. In particular, we find that the phase probability distribution for a NOON state presents equally spaced peaks, whose width depends on the particles number. The sensitivity is at a Heisenberg limit, with

$\beta =0.995\pm 0.005$ and

$\alpha =3.07\pm 0.04$. We are now in a position to compare the sensitivity that can be reached with the different states considered in this work. In

Figure 12 we present the results for the non-interacting case. Here, the sensitivity corresponding to the different analyzed states plotted against the particles number. When the beam splitter is fed with the

$|N,0\rangle $ state, the sensitivity scales as

$\frac{\alpha}{\sqrt{N}}$, close to the-shot noise limit. If, instead, a TF or NOON state is created after the beam splitter,

$\Delta \varphi =\frac{\alpha}{N}$ with the coefficient

${\alpha}_{NOON}<{\alpha}_{TF}$. The discussion of the non-interacting limit is useful when reference is considered to the possibility of adding inter-well interactions. Indeed, as we discussed in this section, introducing inter-well interactions can be equivalent to be within non-interacting conditions. As a result, this work demonstrates that adding the non-local term may compensate this degradation and restore the best scaling and sensitivity.