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We theoretically analyse a multi-modes atomic interferometer consisting of a sequence of Kapitza-Dirac pulses (KD) applied to cold atoms trapped in a harmonic trap. The pulses spatially split the atomic wave-functions while the harmonic trap coherently recombines all modes by acting as a coherent spatial mirror. The phase shifts accumulated among different KD pulses are estimated by measuring the number of atoms in each output mode or by fitting the density profile. The sensitivity is rigorously calculated by the Fisher information and the Cramér-Rao lower bound. We predict, with typical experimental parameters, a temperature independent sensitivity which, in the case of the measurement of the gravitational constant g can significantly exceed the sensitivity of current atomic interferometers.

The goal of interferometry is to estimate the unknown value of a phase shift. The phase shift can arise because of a difference in length among two interferometric arms, as in the first optical Michelson-Morley probing the existence of aether or in LIGO and VIRGO gravitational wave detectors [

Since the last decade, matter wave interferometers have progressively become very competitive when measuring electromagnetic or inertial forces. In particular, atom interferometers [

The sensitivity of light-pulse atom interferometry scales linearly with the space-time area enclosed by the interfering atoms. Large-momentum-transfer (LMT) beam splitters have been suggested [

As an alternative to the atomic fountains, where the atoms follow ballistic trajectories, the interferometric operations can be implemented with trapped clouds [

The initial configuration of the interferometer is provided by a cloud of cold atoms trapped by an harmonic potential

i) Beam-splitter: A KD pulse is applied to the atomic cloud state at the time

ii) Phase shift: Each spatial mode gains a phase shift

iii) Beam splitter: the harmonic trap coherently recombines the wave packets and a second KD pulse is applied to again mix and separate the modes along different paths.

iv) Measurement: The phase shift is estimated by fitting the atomic density profile or by counting the number of atoms in each spatial mode at

The sequences i)-iii) can be iterated an arbitrary number of times n before the final measurement iv).

The plan of the paper is as follows. In Section 2, we present a detailed description of the multi-modes KD interferometer. As an application we calculate the Fisher information and the Cramér-Rao lower bound sensitivity [

Let’s consider first a single atom described by a wave packet

where

with

where we have used the Bessel generating function

After the application of the first KD, the wave-packets are coherently driven by the harmonic trap and recombined after a time

Furthermore, in presence of an external field, each spatial mode created by the KD beam splitter gains during the time

After iterating a number of times n the sequence of KD pulses and phase shift accumulations, the wave function at

where

and

while for even n we have

After n iterations, a last KD pulse is applied at the time

The wave packets gain their maximum spatial separation after a further

Eventually, the wave function at the final time

where

and

with

In the limit if zero overlap between the various wave packets in Equation (8),

the density function at the measurement time

Equation (12) shows that there are

In the limit of a large number

where N is the number of uncorrelated atoms. F denotes the Fisher information calculated from the particle density at the measurement time

With Equation (12), Equation (14) becomes

(see Appendix). We finally obtain

where

Notice that even in the case of a odd value of n, with

which can also be written as

since the total number of modes is

We remark here the important condition of non overlap of the wave packets corresponding to the different momentum modes at the time of measurement, Equation (11). A further interesting point is that Equation (18) is independent from the temperature of the atoms as long as their de Broglie wavelength remains larger than the internal spatial separation of the periodic potential creating the Kapitza-Dirac pulse. We will show this in the following Sections by considering as a specific application the interferometric estimation of the gravitational constant.

We now investigate the KD interferometer theory to estimate the gravity constant g. The evolution of the initial state

To take in account the effect of the gravitational force on the dynamical evolution of the trapped atom states, we need to include in the free propagator

where

since the quantum propagator undergone with gravity field is reduced to

As expected, each spatial mode gains its phase shift

where

and for even n

The last KD pulse is applied on the wave function Equation (22) at time

Firstly, we consider the case without the gravity field. Then at the time

where

and

where

where

Except the phase difference between Equation (24) and Equation (27), a constant difference d is found in the centre position of each sub-wave packets induced by the gravity field. In the case of “no-overlap” condition (Equation (11)), which is satisfied when the width of the initial wave packet is much larger than the interwell distance of the KD optical lattice (

from Equation (24) or

from Equations ((27), (12) and (28)) show that the information on the estimated values of

We now consider an atomic gas at finite temperature T. To get some simple insight on the physics of the problem, we consider the system as made by a swarm of minimum uncertainty Gaussian wave packets

where the initial wave packet width

Each wav packet evolves driven by the propagators calculated in the previous Section:

and

where

where

It is interesting to note that

or

Notice that the value of the gravitational constant g is only contained in the weights of the modes.

The requirement is that sub-wave packets in Equation (37) are spatially separated, which means

which means

As expected, the spatial separation condition in Equation (37)

Substituting the density function Equation (37) at the measurement time

The Fisher information for our system depends on the temperature, initial density profile, the interferometer transformation, and the choice of the observable that, here, is the spatial position of atoms. In this case, the estimator can simply be a fit of the final density profile. However, the same results would be obtained by choosing as observable, the number of particles in each Gassian spatial mode. Since the initial state is made of uncorrelated atoms, there is no need to measure correlations between the modes in order to saturate the Cramér-Rao lower bound Equation (13) at the optimal value of the value phase shift.

Before proceeding to discuss the finite temperature case, we calculate the highest sensitivity of the unbiased estimation of parameter g, which is guaranteed by the no-overlap condition

In the limit

where

The Equation (43) can be rewritten as

where

If the gravity field is witched on in the last KD pulse, the density profile at final time is described by Equation (38). In this case, there is a further contribution to the Fisher Equation (42) from the shift on the center of sub-wave packets and we have

We now estimate the expected sensitivity under realistic experiment conditions. We consider 10^{5} ^{88}Sr atoms trapped in an harmonic trap having

With a strength of the KD potential ^{88}Sr atoms is also increased from

Since the thermal de Broglie wavelength decreases when increasing the temperature, the no-overlap condition Equation (11) breaks down at

As a comparison with current atom interferometers, we calculate the sensitivity obtained from a simple interference pattern observed after a free expansion of an initial atom clouds relevant, for instance, when measuring the gravitational constant g using Bloch oscillations [

where

where

where

where

where we have estimated the maximum occupied lattice sites by

Considering the sensitivity for a single Kaptiza-Dirac pulse with Equation (51), we can reach a sensitivity larger than 3 order of magnitude that the sensitivity obtained in an interference pattern. The reason is that the KD pulses can create several wave packets spanning a distance^{3} with only once KD pulse and typical values of the experimental parameters. A further advantage is that such high sensitivity interferometry can be realised with a compact experimental setup.

We now consider the effects of noise and imperfections on the sensitivity of the interferometer. We mainly consider two kinds of perturbations, which may arise from the experimental realization of the interferometry. The first one is the effect of the anharmonicity, described by a position dependent random perturbation, and the second one the effect of a shift in position between different sequences of the KD pulses.

The effect of anharmonicity is investigated by numerically simulating the interferometric sequences with the following potential

where

Starting with the ground state of the harmonic trap

the average density

information

Due to the perturbation potential

A shift of the optical lattice with respect to the harmonic trap

tive KD pulses

where

where

makes only an phase shifts for each sub-wave packets. Therefore, the non-overlap condition Equation (11) does not have any modification even after considering the off-center shift In this case, the final density profiles is

Equation (56) shows that the center shifts could induce a fluctuation by

During the last few decades, matter-wave interferometry has been successfully extended to the domain of atoms and molecules. Most current interferometric protocols for the measurement of gravity or inertial forces are based on the manipulation of free falling atoms realizing Mach-Zehnder like configurations. Here we propose an atomic multimode interferometer with atoms trapped in a harmonic potential and where the multi beam-splitter operation are implemented with Kapitza-Dirac pulses. The mirror operations are performed by the harmonic trap which coherently drives a tunable number of spatially addressable atomic beams. All interferometer processes, including splitting, phase accumulation and reflection are performed and completed within the harmonic trap. Therefore, all trapped atoms contribute to the sensitivity. We have applied our scheme to the estimation of the gravitational constant and estimate, with realistic experimental parameters, a sensitivity of 10^{−9}, significantly exceeding the sensitivity of current interferometric protocols.

Our work is supported by the National Science Foundation of China (No. 11374197), PCSIRT (No. IRT13076), the National Science Foundation of China (No.11504215).

Cheng, R.L., He, T.C., Li, W.D. and Smerzi, A. (2016) Theory of a Kaptiza-Dirac Interferometer with Cold Trapped Atoms. Journal of Modern Physics, 7, 2043-2062. http://dx.doi.org/10.4236/jmp.2016.715180

To obtain Equation (15), we have considered the Bessel functions identity

With this we have

where one more identity

has been used to obtain

For Equation (45), Using the Equation (41) and Equation (29), we obtain

By using the initial state

we get

where

The second step uses the “no-overlap” condition by changing

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