Contrast and phase-shift of a trapped atom interferometer using a thermal ensemble with internal state labelling

A very convenient rule of thumb to infer the coherence time can be derived from equation (5) by considering the second order Taylor expansion of C under the assumption $| \delta {\omega }_{n}| t\ll 1$. This leads to $C(t)\simeq 1-{(t/{t}_{c})}^{2}/2$, where t_c is understood as the coherence time, with the following expression for t_c:

$$\begin{eqnarray}&&{t}_{c}\simeq {\left[\displaystyle \sum _{n}{p}_{n}\delta {\omega }_{n}^{2}-{\left(\displaystyle \sum _{n}{p}_{n}\delta {\omega }_{n}\right)}^{2}\right]}^{-1/2}.\end{eqnarray} \tag{ 7 }$$

In other words, the inferred decoherence rate ${t}_{c}^{-1}$ is on the same order of magnitude as the standard deviation of the $\delta {\omega }_{n}$, weighted by the Boltzmann factors p_n.

If we furthermore assume that V_a and V_b correspond, during the interrogation period, to two harmonic traps with slightly different frequencies ${\omega }_{a}$ and ${\omega }_{b}$, with $| {\omega }_{a}-{\omega }_{b}| \ll {\omega }_{a,b}$, equation (7) leads, in the case of a weakly degenerate gas ${\hslash }{\omega }_{a,b}\ll {kT}$, to:

$$\begin{eqnarray}&&{t}_{c}\simeq \displaystyle \frac{1}{\delta \omega }\displaystyle \frac{{\hslash }\omega }{{kT}},\end{eqnarray} \tag{ 8 }$$

with $\omega =({\omega }_{a}+{\omega }_{b})/2$ and $\delta \omega =| {\omega }_{a}-{\omega }_{b}|$. It is obvious from equation (8) that t_c increases with symmetry and decreases with temperature, as expected intuitively. This result differs from the exact calculation, in case of two harmonic potentials [8], only by a factor $\sqrt{3}$. For a typical temperature of 500 nK, equation (8) gives a symmetry-limited coherence time on the order of 15 ms for a realistic value of the asymmetry $\delta \omega /\omega \,\lesssim \,$ 10⁻³ [17]. In the case of non-harmonic traps, equations (5) or (7) can still be used with perturbatively—or numerically—estimated values of the eigen-energies.

In the rest of this paper, we consider the case where ${V}_{i}(\widehat{z})$ is the sum of a harmonic potential and an acceleration or a gravity potential namely:

$$\begin{eqnarray}\begin{array}{rcl}{V}_{i}(\widehat{z}) & = & \displaystyle \frac{m{\omega }_{i}^{2}}{2}{(\widehat{z}-{z}_{i})}^{2}+{mg}\widehat{z}\\ & = & \displaystyle \frac{m{\omega }_{i}^{2}}{2}{(\widehat{z}-{z}_{i}^{{\rm{cm}}})}^{2}+\displaystyle \frac{{{mg}}^{2}}{2{\omega }_{i}^{2}}+{{mgz}}_{i}^{{\rm{cm}}},\end{array}\end{eqnarray} \tag{ 9 }$$

where m is the atomic mass, ${\omega }_{i}$ are the trap frequencies, g is the acceleration or gravity field, z_i is the trap centre (minimum of the trapping part of the potential) and ${z}_{i}^{{\rm{cm}}}={z}_{i}-g/{\omega }_{i}^{2}$ is the centre of mass position of the atoms. The phase difference ${\rm{\Delta }}\varphi (t)$ (equation (6)) after an interrogation time t, stemming from Hamiltonian (1) and potential (9), is given in this case by:

$$\begin{eqnarray}&&{\rm{\Delta }}\varphi (t)=(\omega -{\omega }_{{ab}})t-{\rm{\Delta }}{\varphi }^{0}(t),\end{eqnarray} \tag{ 10 }$$

with:

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{\varphi }^{0}(t) & = & \left[{mg}({z}_{b}^{{\rm{cm}}}-{z}_{a}^{{\rm{cm}}})+\displaystyle \frac{{{mg}}^{2}}{2}\left(\displaystyle \frac{1}{{\omega }_{b}^{2}}-\displaystyle \frac{1}{{\omega }_{a}^{2}}\right)\right]\displaystyle \frac{t}{{\hslash }}\\ & & +\displaystyle \frac{{\omega }_{b}-{\omega }_{a}}{2}t+{\phi }_{T}(t),\end{array}\end{eqnarray} \tag{ 11 }$$

where :

$$\begin{eqnarray}&&{\phi }_{T}(t)=\arctan \left\{\displaystyle \frac{\sin (({\omega }_{b}-{\omega }_{a})t){{\rm{e}}}^{-{\hslash }{\omega }_{a}/({kT})}}{1-\cos (({\omega }_{b}-{\omega }_{a})t){{\rm{e}}}^{-{\hslash }{\omega }_{a}/({kT})}}\right\}.\end{eqnarray} \tag{ 12 }$$

In equation (10) ${\rm{\Delta }}{\varphi }^{0}(t)$ arises from the spatial separation of the two internal states, and $(\omega -{\omega }_{{ab}})t$ describes the free evolution of the states. In equation (11), the first term is the classical difference in potential energy due to the presence of the acceleration or gravity field. The second term is an energy shift resulting from the addition of the harmonic potential with the linear ${mg}\widehat{z}$ term (see equation (9)). The third term is the difference of zero point energies of the two harmonic oscillators. The last term, which is temperature dependent, vanishes in two cases: (i) a symmetric interferometer (i.e. ${\omega }_{a}={\omega }_{b}$), (ii) zero-temperature. Equation (12) shows that not only the contrast depends on temperature (as was predicted in [8]) but also the phase-shift. We also predict a direct link between the phase-shift and the relative asymmetry of the two traps, as was previously pointed out in [18].

It has been demonstrated in [12, 13] that non-trivial temporal ramps can be used to move an atomic cloud while keeping the population of the different quantum levels unchanged at the ends of the ramp, on the time scale of the trapping period (hence much faster than an adiabatic ramp [8]). Such ramp can also be used to move trapped cold ions [20–22] while keeping their initial state at the end of the ramp. We propose, in the following, to apply this technique, known as STA [10, 12–14], to the case of a trapped thermal atom interferometer. For simplicity, we only consider the case of a harmonic trap (for other potentials the reader is referred to [10] and references therein). We thus consider a trapping potential with a time-depend position and stiffness:

$$\begin{eqnarray}&&{V}_{i}(\widehat{z},t)=\displaystyle \frac{m{\omega }_{i}^{2}(t)}{2}{[\widehat{z}-{z}_{i}(t)]}^{2}+{mg}\widehat{z}.\end{eqnarray} \tag{ 13 }$$

Similar to the case of equation (9), we can rewrite these potentials as:

$$\begin{eqnarray}{V}_{i}(\widehat{z},t) & = & \displaystyle \frac{m{\omega }_{i}^{2}(t)}{2}{\left[\widehat{z}-{z}_{i}(t)+\displaystyle \frac{g}{{\omega }_{i}^{2}(t)}\right]}^{2}+{\gamma }_{i}(t)\\ \mathrm{with}\ :{\gamma }_{i}(t) & = & -\displaystyle \frac{{{mg}}^{2}}{2{\omega }_{i}^{2}(t)}+{{mgz}}_{i}(t).\end{eqnarray} \tag{ 14 }$$

To introduce the mathematical condition which must be fulfilled for the STA, we need to write a dynamical invariant ${\widehat{I}}_{i}(t)$ of $\widehat{H}| i\rangle \langle i| (t)$. $\widehat{K}$ is a dynamical invariant of an operator $\widehat{P}$ if [23]: (i) $j{\hslash }{\partial }_{t}\widehat{K}+[\widehat{K},\widehat{P}]=0$ and (ii) $\widehat{K}$ is hermitian. Expressions for ${\widehat{I}}_{i}(t)$ can be found in the literature [12, 24, 25]. After adapting them to include the presence of g, we obtain:

$$\begin{eqnarray}&&{\widehat{I}}_{i}(t)=\displaystyle \frac{{\omega }_{0}}{2m}{[{\rho }_{i}(\widehat{p}-m{\dot{z}}_{i}^{{\rm{cm}}})-m{\dot{\rho }}_{i}(\widehat{z}-{z}_{i}^{{\rm{cm}}})]}^{2}+\displaystyle \frac{m{\omega }_{0}}{2}\displaystyle \frac{{(\widehat{z}-{z}_{i}^{{\rm{cm}}})}^{2}}{{\rho }_{i}^{2}},\end{eqnarray} \tag{ 15 }$$

where ${\omega }_{0}$ is an arbitrary angular frequency and ${\rho }_{i}$ and z_i^cm are solutions of the following equations:

$$\begin{eqnarray}&&{\ddot{\rho }}_{i}+{\omega }_{i}^{2}(t){\rho }_{i}=\displaystyle \frac{1}{{\rho }_{i}^{3}},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{\ddot{z}}_{i}^{{\rm{cm}}}+{\omega }_{i}^{2}(t)\left[{z}_{i}^{{\rm{cm}}}-{z}_{i}(t)+\displaystyle \frac{g}{{\omega }_{i}^{2}(t)}\right]=0.\end{eqnarray} \tag{ 17 }$$

Equation (16) is the Ermakov equation and equation (17) is the classical linear oscillator. Physically, z_i^cm is the centre of mass of the atomic cloud obeying equation (17), and ${\rho }_{i}$ is proportional to the cloud size [12]. For a given time $t={t}_{p}$, the populations of the different quantum levels will be the same as for $t={t}_{m}$ if $\widehat{H}| i\rangle \langle i| ({t}_{m})\propto \widehat{{I}_{i}}({t}_{m})$ and $\widehat{H}| i\rangle \langle i| ({t}_{p})\propto \widehat{{I}_{i}}({t}_{p})$ [10, 12]. This imposes in particular the following conditions on ${\rho }_{i}$ and z_i^cm at ${t}_{m,p}$:

$$\begin{eqnarray}{\rho }_{i}({t}_{m,p}) & = & \displaystyle \frac{1}{\sqrt{{\omega }_{i}({t}_{m,p})}},\quad {z}_{i}^{{\rm{cm}}}({t}_{m,p})={z}_{i}({t}_{m,p})-\displaystyle \frac{g}{{\omega }_{i}^{2}({t}_{m,p})},\\ {\dot{\rho }}_{i}({t}_{m,p}) & = & 0,\qquad \qquad {\dot{z}}_{i}^{{\rm{cm}}}({t}_{m,p})=0,\end{eqnarray} \tag{ 18 }$$

where ${\omega }_{i}({t}_{m,p})$ and ${z}_{i}({t}_{m,p})$ are fixed parameters which are linked to the equilibrium position and cloud size at ${t}_{m,p}$. Two additional conditions: ${\ddot{\rho }}_{i}({t}_{m,p})=0$ and ${\ddot{z}}_{i}^{{\rm{cm}}}({t}_{m,p})=0$ are provided by (16) and (17). Together with (18) they form the STA conditions at ${t}_{m,p}$.

In order to find a temporal ramp on ${\omega }_{i}$ and z_i for the splitting, we need to solve equations (16), (17) and (18). To do this, as we have six conditions on ${\rho }_{i}$ and six on z_i^cm, we take a fifth-order polynomial ansatz for ${\rho }_{i}$ and z_i^cm [12–14]. The frequency ramp is first found from ${\rho }_{i}$ and (16) and the trap position z_i is then deduced from ${\omega }_{i}$, z_i^cm and (17). To give a numerical example, the following parameters are taken (times are defined in figure 2): ${t}_{1}=2\,\mathrm{ms}$, ${\omega }_{i}(0)/2\pi =1\,\mathrm{kHz}$, ${\omega }_{i}({t}_{1})/2\pi =500\,\mathrm{Hz}$, g is the gravitational acceleration and the maximum separation distance between the two internal states is 200 μm. This example is shown in figure 2, where we use the same ramp for recombination and splitting. Numerically we were not able to find t₁ significantly lower than 2 ms while preserving a smooth ramp for the frequency (without imaginary frequencies to keep the trapping behaviour of the potential) and for the trap position. This is in accordance with [26] where it is stated that the minimum time is on the order of $2\pi /{\omega }_{i}$.

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For purposes of interferometry, the contribution to the overall phase shift of the splitting and merging period has to be taken into account, all the more since their duration is not negligible compared to the typical value of the coherence time inferred previously. The framework of the dynamical invariant $\widehat{{I}_{i}}(t)$ [23] provides a tool to compute this overall phase shift between t = 0 and $t={t}_{f}$ (i.e. during the whole interferometric sequence). Reference [23] gives the following generic solution $| t\rangle | i\rangle$ of the Schrödinger equation with a time-dependent hamiltonian $\widehat{H}| i\rangle \langle i| (t)$:

$$\begin{eqnarray}&&| t\rangle | i\rangle =\displaystyle \sum _{n}\,{c}_{n}^{i}\exp (j{\alpha }_{n}^{i}(t))| n(t)\rangle | i\rangle ,\end{eqnarray} \tag{ 19 }$$

where c_nⁱ are time-independent factors which depend on the initial conditions, $| n(t)\rangle | i\rangle$ are the eigen-states of ${\widehat{I}}_{i}(t)$ and the ${\alpha }_{n}^{i}(t)$ are chosen such that $\exp (j{\alpha }_{n}^{i}(t))| n(t)\rangle | i\rangle$ are solutions of the Schrödinger equation with the hamiltonian $\widehat{H}| i\rangle \langle i| (t)$ [23]. Adapting the results of [12, 27, 28] to the case of the trapped interferometer considered in this paper, we obtain:

$$\begin{eqnarray}\arg (\exp (j{\alpha }_{n}^{i}(t))| n(t)\rangle | i\rangle ) & = & -\left(n+\displaystyle \frac{1}{2}\right){\displaystyle \int }_{0}^{t}\displaystyle \frac{{\rm{d}}{t}^{\prime} }{{\rho }_{i}^{2}(t^{\prime} )}+{{\rm{\Psi }}}_{i}(z,t)\\ & & -\displaystyle \frac{{F}_{i}(t)}{{\hslash }}-\displaystyle \frac{{{\rm{\Gamma }}}_{i}(t)}{{\hslash }},\end{eqnarray} \tag{ 20 }$$

with the following expressions for ${{\rm{\Psi }}}_{i}$, F_i and ${{\rm{\Gamma }}}_{i}$:

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Psi }}}_{i}(z,t) & = & \displaystyle \frac{m}{{\hslash }}\left[\displaystyle \frac{\dot{{\rho }_{i}}}{2{\rho }_{i}}{z}^{2}+\displaystyle \frac{1}{{\rho }_{i}}({\dot{z}}_{i}^{{\rm{cm}}}{\rho }_{i}-{z}_{i}^{{\rm{cm}}}{\dot{\rho }}_{i})z\right]\\ {F}_{i}(t) & = & \displaystyle \frac{m}{2}{\displaystyle \int }_{0}^{t}{\rm{d}}{t}^{\prime} \left[\displaystyle \frac{1}{{\rho }_{i}^{2}}{({\dot{z}}_{i}^{{\rm{cm}}}{\rho }_{i}-{z}_{i}^{{\rm{cm}}}{\dot{\rho }}_{i})}^{2}\right]\\ & & +\displaystyle \frac{m}{2}{\displaystyle \int }_{0}^{t}{\rm{d}}{t}^{\prime} \left[-\displaystyle \frac{{({z}_{i}^{{\rm{cm}}})}^{2}}{{\rho }_{i}^{4}}+{\omega }_{i}^{2}{\left({z}_{i}-\displaystyle \frac{g}{{\omega }_{i}^{2}}\right)}^{2}\right]\\ {{\rm{\Gamma }}}_{i}(t) & = & \displaystyle \frac{m}{2}{\displaystyle \int }_{0}^{t}{\rm{d}}{t}^{\prime} \left[2{{gz}}_{i}-\displaystyle \frac{{g}^{2}}{{\omega }_{i}^{2}}\right].\end{array}\end{eqnarray} \tag{ 21 }$$

In the potential (13) the linear term ${mg}\widehat{z}$ shifts the energy by a quantity ${\gamma }_{i}(t)$ which corresponds to ${{\rm{\Gamma }}}_{i}(t)$ and translates the centre of the harmonic term by a distance $-g/{\omega }_{i}^{2}$ which is accounted for in ${{\rm{\Gamma }}}_{i}(t)$. In a perfect adiabatic case only the first term of equation (20) and ${{\rm{\Gamma }}}_{i}(t)$ remain.

As the $\exp (j{\alpha }_{n}^{i}(t))| n(t)\rangle | i\rangle$ are all solutions of the Schrödinger equation with the Hamiltonian $\widehat{H}| i\rangle \langle i| (t)$ and form a orthonormal basis of our Hilbert space [12, 23, 27, 28], we can easily extend equation (3) to account for time-dependent splitting and recombination. Thus from (20) and (21) we can compute the contrast and the phase-shift. In this case the term ${{\rm{\Omega }}}_{n}^{b}-{{\rm{\Omega }}}_{n}^{a}$ from the definition of A(t) is equal to: $\arg (\exp (j{\alpha }_{n}^{b}(t))| n(t)\rangle | b\rangle )-\arg (\exp (j{\alpha }_{n}^{a}(t))| n(t)\rangle | a\rangle )$. Under the same hypothesis as in the adiabatic case (equation (8)), the coherence time t_c can be inferred by solving the following equation:

$$\begin{eqnarray}&&\sqrt{3}=\displaystyle \frac{{kT}}{{\hslash }\omega }\left|{\int }_{0}^{{t}_{c}}\left(\displaystyle \frac{1}{{\rho }_{a}^{2}}-\displaystyle \frac{1}{{\rho }_{b}^{2}}\right){\rm{d}}{t}\right|,\end{eqnarray} \tag{ 22 }$$

which is a dynamical version of equation (8) for time-dependent frequencies. It is interesting to notice that z_i(t) has no role in this expression, which is consistent with the fact that a translation or a rotation of an Hamiltonian preserves its eigen-values, and thus it preserves the contrast as already pointed out in [8].

As regards the phase-shift ${\rm{\Delta }}\varphi (t)$, only the splitting dependent part ${\rm{\Delta }}{\varphi }^{0}(t)$ changes and it is given by: ${\rm{\Delta }}{\varphi }^{0}(t)={{\rm{\Psi }}}_{a}(z,t)-{{\rm{\Psi }}}_{b}(z,t)-{F}_{a}(t)/{\hslash }+{F}_{b}(t)/{\hslash }-{{\rm{\Gamma }}}_{a}(t)/{\hslash }+{{\rm{\Gamma }}}_{b}(t)/{\hslash }-\tfrac{1}{2}f(t)+\arg \left[{\sum }_{n}{p}_{n}\exp (-{jnf}(t))\right]$ with $f(t)={\int }_{0}^{t}1/{\rho }_{a}^{2}{\rm{d}}{t}^{\prime} -{\int }_{0}^{t}1/{\rho }_{b}^{2}{\rm{d}}{t}^{\prime}$. Assuming that STA conditions are fulfilled at t = 0 and $t={t}_{f}$ (only the conditions ${\dot{\rho }}_{a,b}(0)={\dot{\rho }}_{a,b}({t}_{f})=0$ and ${\dot{z}}_{a,b}^{{\rm{cm}}}(0)={\dot{z}}_{a,b}^{{\rm{cm}}}({t}_{f})=0$ are needed), we obtain the following (more explicit) expression for the phase-shift : ${\rm{\Delta }}\varphi ({t}_{f})=(\omega -{\omega }_{{ab}}){t}_{f}-{\rm{\Delta }}{\varphi }^{0}({t}_{f})$:

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{\varphi }^{0}({t}_{f}) & = & \displaystyle \frac{m}{2{\hslash }}{\displaystyle \int }_{0}^{{t}_{f}}[{({\dot{z}}_{a}^{{\rm{cm}}})}^{2}-{({\dot{z}}_{b}^{{\rm{cm}}})}^{2}]{\rm{d}}{t}\\ & & -\displaystyle \frac{{mg}}{{\hslash }}{\displaystyle \int }_{0}^{{t}_{f}}({z}_{a}^{{\rm{cm}}}-{z}_{b}^{{\rm{cm}}}){\rm{d}}{t}\\ & & -\displaystyle \frac{m}{2{\hslash }}{\displaystyle \int }_{0}^{{t}_{f}}\left[{\left(\displaystyle \frac{{\ddot{z}}_{a}^{{\rm{cm}}}+g}{{\omega }_{a}}\right)}^{2}-{\left(\displaystyle \frac{{\ddot{z}}_{b}^{{\rm{cm}}}+g}{{\omega }_{b}}\right)}^{2}\right]{\rm{d}}{t}\\ & & -\displaystyle \frac{1}{2}{\displaystyle \int }_{0}^{{t}_{f}}\left(\displaystyle \frac{1}{{\rho }_{a}^{2}}-\displaystyle \frac{1}{{\rho }_{b}^{2}}\right){\rm{d}}{t}-{\phi }_{T}({t}_{f}),\end{array}\end{eqnarray} \tag{ 23 }$$

where :

$$\begin{eqnarray}&&{\phi }_{T}({t}_{f})=\arctan \left\{\displaystyle \frac{\sin (f({t}_{f})){{\rm{e}}}^{-{\hslash }{\omega }_{a}/({kT})}}{1-\cos (f({t}_{f})){{\rm{e}}}^{-{\hslash }{\omega }_{a}/({kT})}}\right\}.\end{eqnarray} \tag{ 24 }$$

In equation (23), the first term comes from kinetic energy. The second is the classical difference in potential acceleration energy. The third comes from the energy shift of the harmonic oscillator levels in the presence of the overall acceleration field of the atomic cloud $g+{\ddot{z}}_{i}^{{\rm{cm}}}$ (i.e. acceleration of the whole interferometer and acceleration of the trap). The fourth term comes from the difference in zero point energies of the two oscillators. To make the latter more explicit, we point out that in the case where ${\omega }_{i}$ is time-independent, then $1/{\rho }_{i}^{2}={\omega }_{i}$ and the fourth term of equation (23) becomes identical to the third term of equation (11). The last term includes the temperature dependence of the phase shift and it is the analogue of (12) for the time dependent case.

Having identified a promising protocol, one might enquire about its robustness. That is, how accurately must a laboratory apparatus reproduce the calculated parameters such as those in figure 2. A full analysis would require a complete numerical study, which is outside the scope of this paper. We can however, make some general statements. The parameters appearing in equation (23) also appear in various guises in the adiabatic procedure. Roughly speaking, the stability requirements for the STA procedure should not be much more stringent than for adiabatic splitting. For example, consider the adiabatic case where in equation (23) ${\dot{z}}_{i}^{{\rm{cm}}}$ and ${\ddot{z}}_{i}^{{\rm{cm}}}$ are equal to zero. To ensure a 1% stability of the phase shift one must ensure a 1% stability of the trap position and stiffness. In the STA case, if we also suppose a 1% stability on the trap stiffness and position, this gives the same order of magnitude for the stability of the cloud velocity (${\dot{z}}_{i}^{{\rm{cm}}}\sim d/\tau$), acceleration (${\ddot{z}}_{i}^{{\rm{cm}}}\sim d/{\tau }^{2}$) and the time-dependent-stiffness ($1/{\rho }_{i}^{2}\sim {\omega }_{i}$), where τ is the duration of the splitting/recombination ramp and d the splitting distance. We thus expect to achieve a stability of order 1% in the STA case as well.

In a practical implementation of this interferometer [8], the experimental parameters are ${\omega }_{i}$ and z_i, and not ${\rho }_{i}$ and z_i^cm. From the two STA ramps for ${\rho }_{i}$ and z_i^cm, we need to compute the ramps for the two experimental parameters ${\omega }_{i}$ and z_i. In the general case, the computation of z_i requires the knowledge of g, which is the parameter we want to measure. This circle can be broken in the two following cases:

(i) We choose the splitting time t₁ and the trap frequency ${\omega }_{a}={\omega }_{b}=\omega$ such that ${z}_{i}^{{\rm{cm}}}\simeq {z}_{i}$. If we call d the splitting distance, the latter choice and equation (17) imply that ${t}_{1}^{2}{\omega }^{2}\gg 1$ and $g/({\omega }^{2}d)\ll 1$, i.e. an adiabatic splitting and a strong trap confinement to make the acceleration shift of the trap position negligible. In this ideal adiabatic case, the phase-shift ${\rm{\Delta }}{\varphi }^{0}({t}_{f})$ reduces to:

$$\begin{eqnarray}&&{\rm{\Delta }}{\varphi }^{0}({t}_{f})=-\displaystyle \frac{{mg}}{{\hslash }}{\int }_{0}^{{t}_{f}}({z}_{a}-{z}_{b}){\rm{d}}{t},\end{eqnarray} \tag{ 25 }$$

making such a system an attractive candidate for acceleration measurements. Assuming a phase measurement limited by the quantum projection noise leads to an uncertainty on the measurement of g on the order of $\delta g/g\sim {\hslash }/m{\rm{\Delta }}{{zt}}_{c}\sqrt{N}$ per shot. For example, with the following numerical values: ${\rm{\Delta }}z\,\sim \,$ 100 μm, ${t}_{c}\sim 10\,\mathrm{ms}$, $N\sim 1000$ atoms and m = 1.4 × 10⁻²⁵ kg for ${}^{87}$Rb we obtain $\delta g/g\,=\,$ 2 × 10${}^{-6}$ per shot.

(ii) If the adiabatic approximation is not valid for example because of a too short coherence time, it is still possible to use the previously described interferometer to measure an acceleration. In the case of identical time-dependent-stiffness for the two traps, i.e. ${\rho }_{a}={\rho }_{b}$, we suppose that a time-dependent function h exists and satisfies the two following conditions: (1) ${z}_{a}^{{\rm{cm}}}=(d-g/{\omega }_{s}^{2}+g/{\omega }_{r}^{2})h-g/{\omega }_{r}^{2}$ and ${z}_{b}^{{\rm{cm}}}=(-d-g/{\omega }_{s}^{2}+g/{\omega }_{r}^{2})h-g/{\omega }_{r}^{2}$ where ${\omega }_{r}=\omega (0)=\omega ({t}_{f})$, ${\omega }_{s}=\omega ({t}_{1})=\omega ({t}_{2})$ and $d=| {z}_{a}({t}_{1},{t}_{2})| \,=| {z}_{b}({t}_{1},{t}_{2})|$ and (2) the STA conditions are fulfil for z_a^cm and z_b^cm. The important point is that finding such a function h does not imply the knowledge of the acceleration g. In this case, the time dependent-splitting distance is ${z}_{a}-{z}_{b}=2d\ddot{h}/{\omega }^{2}+2{dh}$ and this last function can be used in the interferometer sequence to measure the acceleration g.