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arXiv:1701.04629 · Journal of the Optical Society of America B

Phase shift formulation for N-light-pulse atom interferometers: application to inertial sensing

Malo Cadoret, Nassim Zahzam, Yannick Bidel, Clément Diboune, Alexis Bonnin, Fabien Théron, Alexandre Bresson

Published Jul 25, 2016• 11 citations

Abstract

We report on an original and simple formulation of the phase shift in N-light-pulse atom interferometers. We consider atomic interferometers based on two-photon transitions (Raman transitions or Bragg pulses). Starting from the exact analytical phase shift formula obtained from the atom optics ABCD formalism, we use a power series expansion in time of the position of the atomic wave packet with respect to the initial condition. The result of this expansion leads to a formulation of the interferometer phase shift where the leading coefficient in the phase terms up to T^k dependences (k >= 0) in the time separation T between pulses, can be simply expressed in terms of a product between a Vandermonde matrix, and a vector characterizing the two-photon pulse sequence of the interferometer. This simple coefficient dependence of the phase shift reflects very well the atom interferometer's sensitivity to a specific inertial field in the presence of multiple gravito-inertial effects. Consequently,we show that this formulation is well suited when looking for selective atomic sensors of accelerations, rotations, or photon recoil only, which can be obtained by simply zeroing some specific coefficients. We give a theoritical application of our formulation to the photon recoil measurement.

DOI: 10.1364/josab.33.001777

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We report on an original and simple formulation of the phase shift in N-light-pulse atom interferometers. We consider atomic interferometers based on two-photon transitions (Raman transitions or Bragg pulses). Starting from the exact analytical phase shift formula obtained from the atom optics ABCD formalism, we use a power series expansion in time of the position of the atomic wave packet with respect to the initial condition. The result of this expansion leads to a formulation of the interferometer phase shift where the leading coefficient in the phase terms up to T^k dependences (k >= 0) in the time separation T between pulses, can be simply expressed in terms of a product between a Vandermonde matrix, and a vector characterizing the two-photon pulse sequence of the interferometer. This simple coefficient dependence of the phase shift reflects very well the atom interferometer's sensitivity to a specific inertial field in the presence of multiple gravito-inertial effects. Consequently,we show that this formulation is well suited when looking for selective atomic sensors of accelerations, rotations, or photon recoil only, which can be obtained by simply zeroing some specific coefficients. We give a theoritical application of our formulation to the photon recoil measurement.

In the framework of the ABCD ξ formulation of atom optics and with an adequate modelization of the beam splitters, we establish an exact analytical phase shift expression for atom interferometers. This result is valid for a time-dependent external Hamiltonian at most quadratic in position and momentum operators and is expressed in terms of coordinates and momenta of the wave packet centres at the interaction vertices only. As a specific application, the case of atom gyrometers and accelerometers is presented in detail.

In the presence of Earth gravity and gravity-gradient forces, centrifugal and Coriolis forces caused by the Earth rotation, the phase of the time-domain atom interferometers is calculated with accuracy up to the terms proportional to the fourth degree of the time separation between pulses. We considered double-loop atom interferometers and found appropriate condition to eliminate their sensitivity to acceleration to get atomic gyroscope, or to eliminate the sensitivity to rotation to increase accuracy of the atomic gravimeter. Consequent use of these interferometers allows one to measure all components of the acceleration and rotation frequency projection on the plane perpendicular to gravity acceleration. Atom interference on the Raman transition driving by noncounterpropagating optical fields is proposed to exclude stimulated echo processes which can affect the accuracy of the atomic gyroscopes. Using noncounterpropagating optical fields allows one to get a new type of the Ramsey fringes arising in the unidirectional Raman pulses and therefore centered at the two-quantum line center. Density matrix in the Wigner representation is used to perform calculations. It is shown that in the time between pulses, in the noninertial frame, for atoms with fully quantized spatial degrees of freedom, this density matrix obeys classical Liouville equations.

High-precision inertial sensing demonstrations with light pulse atom interferometry have typically used Raman pulses having durations orders of magnitude shorter than the dwell time between interferometer pulses. Environmentally robust sensors operating at high-bandwidth will be required to operate at short (millisecond scale) dwell times between Raman pulses. In such an operational mode, the Raman pulse duration becomes an appreciable fraction of the dwell time between pulses. In addition, high-precision inertial sensing applications have typically been demonstrated in mildly dynamic or nondynamic environments having low rate of change of inertial input, ensuring that applied Raman pulses satisfy the Raman resonance condition. Application of nonresonant pulses will be inevitable in sensors registering time-varying inertial input. We present a diagrammatic technique for calculation of atomic output state populations for multipulse atom optics manipulations that explicitly account for the effects of finite pulse duration and finite Raman detuning effects on the laser-induced atomic phase. We analyze several atom interferometer sequences. We report accelerometer and gyroscope phase evolution for fixed Raman laser frequency difference incorporating corrections in powers of the ratio of pulse duration to time interval between interferometer pulses. Our accelerometer result agrees with other published results.

Atom interferometry is a sensitive tool for measuring relativistic effects, but there are no known non-trivial exact solutions for relativistic atom interferometer phase shifts. Here we derive relativistically exact expressions within the usual semiclassical approximation for a wide range of experimentally interesting atom interferometer pulse sequences in flat spacetime, including Mach-Zehnder, resonant, and large momentum transfer interferometer geometries. As an example, the leading order phase shift

ω_a g T^2/c

for a Mach-Zehnder clock atom interferometer is found to become

ω_a \left(1 + \frac{ω_a}{2m}\right)(e^{-gT/c}-1)^2 c/g

when all relativistic kinematics are included. We calculate exact phase shifts for both clock (single-photon) interferometers and Raman or Bragg (two-photon) interferometers.

Summary form only given. Atom interferometry based on cold atoms and light beamsplitters is a mature technique, which allows for the realization of highly sensitive inertial sensors. In the most advanced inertial sensors based on atom interferometry, atomic wavepackets are manipulated by light beamsplitters using two-photon Raman transitions. The interferometer phase shift, which depends on the acceleration and the rotation of the experimental setup with respect to the inertial reference frame defined by the atoms in free fall, is derived from the relative displacement of the atoms with respect to the beamsplitter laser's equiphases. Wavefront aberrations thus induce parasitic phase shifts, which bias the measurement, and which are linked to the residual ballistic motion of the atoms across the laser beam profile during their free fall and cancels out at zero atomic temperature. The wavefront distortions of the lasers beamsplitters is an important limit in the accuracy of atom interferometers and in particular, the dominating contribution in best cold atom gravimeters (CAG) uncertainty budget. This work clears the way towards unprecedented accuracies for absolute gravimetry, below the 1 μGal level. The method proposed can be applied to any atomic sensor based on light beamsplitters, which are inevitably affected by distortions of the lasers wavefronts. The improved control of systematics it provides will have significant impact in high precision measurements with atom interferometry, with important applications to geodesy, fundamental physics tests and to the development of highest grade inertial sensors.