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Physical Review A

Pancharatnam phase: A tool for atom optics

Boris Décamps, A. Gauguet, J. Vigué, M Büchner

Published Jul 24, 2017• 8 citations
Publisher

Abstract

The Pancharatnam phase belongs to the family of geometric Berry phases. We use this optical phase to control the phase of our atom interferometer, which involves diffraction of the atom wave by laser standing waves in the Bragg regime. The Pancharatnam phase of the reflected beam of one standing wave controls the phase imprinted on the atom wave by the diffraction process. In addition to the expected phase shift, the experimental data exhibits the signature of several defects which are described and quantified. From this analysis, we estimate that a Pancharatnam phase shifter can be reliably used to control the phase of an atom interferometer in the sub-mrad regime. Moreover, as the geometric nature of the Pancharatnam phase renders this phase achromatic, its use in multispecies atom interferometers may be of great interest.

DOI
10.1103/physreva.96.013624

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The Pancharatnam phase belongs to the family of geometric Berry phases. We use this optical phase to control the phase of our atom interferometer, which involves diffraction of the atom wave by laser standing waves in the Bragg regime. The Pancharatnam phase of the reflected beam of one standing wave controls the phase imprinted on the atom wave by the diffraction process. In addition to the expected phase shift, the experimental data exhibits the signature of several defects which are described and quantified. From this analysis, we estimate that a Pancharatnam phase shifter can be reliably used to control the phase of an atom interferometer in the sub-mrad regime. Moreover, as the geometric nature of the Pancharatnam phase renders this phase achromatic, its use in multispecies atom interferometers may be of great interest.
The geometric phase was first anticipated 1956 by Pancharatnam, in the context of polarised light. Much later, in 1984, Berry made an analogue discovery, but this time in the context of quantum mechanics. It turns our that the geometric phase, or Berry's phase, or the Pancharatnam phase, is a very general phenomenon. It is a phase shift that can be acquired by a system, when it has undergone a adiabatic process, bringing it back to its original initial conditions. If the parameter space of the involved Hamiltonian is curved, the final condition may involve a phase change. This effect has been demonstrated in a range of different interferometry experiments. Although the Pancharatnam phase has been seen, this has always been in studies involving the first order correlation, g(1), this is in the intereferogramm of two or more wave amplitudes. Here, we demonstrate experimental proofs of the effect of the Pancharatnam phase in intensity interferometry, or in other words, where the second order correlation factor g(2) is the relevant description of the interference.
In 1950, Ramsey and Hahn separately described a method using separated oscillatory magnetic fields and electronic levels to measure interference in atomic states. Today, similar physics is exploited to make atom interferometer sensors utilizing what is called the spin-echo sequence. In 1956, Pancharatnam described the geometric phase difference between two polarized states of light. In 1984 Berry described the phase accrued in cyclic and adiabatic quantum systems. A Pancharatnam-Berry phase experiment involving a quarter-wave – half-wave – quarter-wave plate stack of optics and an experiment using the spin-echo sequence are based on seemingly different phenomena in optics versus atomic optics, respectively. However, the mathematical representations used to describe the Pancharatnam-Berry phase in optics are identical to those used to describe the spin-echo sequence. Therefore, the relationship between the spin-echo sequence and the Pancharatnam-Berry phase was explored. Experiments to test the comparisons using a pre-existing cold atom experiment, ~150 μK, were performed. The mathematical connection between the two types of experiments was explained more deeply.
Young's two-beam interference and the Pancharatnam-Berry geometric phase constitute two landmark results in optical physics. Employing the spectral interference law of electromagnetic waves, we establish a general expression for the Pancharatnam-Berry phase in Young's two-pinhole setup. We show that the phase is elegantly specified by the electric field Stokes parameters at the openings. The geometric phase can also be deduced from the interference pattern using Pancharatnam's definition of the generalized phase, as we demonstrate both numerically and experimentally. Our results imply a way to control the phase differences in two-beam electromagnetic interference by adjusting the input-field intensities and polarization states.
Bragg diffraction of matter waves is an established technique used in the most accurate quantum sensors. It is also the method of choice to operate large-momentum-transfer, high-sensitivity atom interferometers. It suffers, however, from an intrinsic multi-path character. Optimal control theory (OCT) has recently led to an improved robustness of atom interferometers to a range of challenging environmental effects such as vibrations or platform accelerations. In this theoretical work, we apply OCT protocols to control the Bragg diffraction phase shifts, thereby enhancing the metrological accuracy of the interferometer. We show a minimization of the diffraction phase for realistic conditions of finite temperature of the incoming wavepacket in a multi-path, high-order Bragg interferometer in a Mach–Zehnder configuration. We study input states with different momentum widths and find that our approach mitigates diffraction phases below the microradian level in the case of 1% of the photon recoil, thereby eliminating one of the leading systematic effects in atom interferometry.
We present a concept for measuring a Berry phase using an atom interferometer in an optical Ramsey arrangement. The adiabatic cyclic process needed for the appearance of a Berry phase consists of the adiabatic change of population in one of the interferometer arms. A closed circuit in parameter space is realized by the time-dependent Doppler detuning of two additional strongly focused Gaussian beams, which couple selectively to one of the interferometer arms. It is shown that the resulting additional atomic phase consists of a Berry phase corresponding to the phase difference of the two Gaussian beams and a dynamical phase caused by the dynamical Stark effect. By recombining the two interferometer arms this total phase gives rise to a shift of the interference pattern. In case of a Ramsey arrangement with magnesium atoms the parameters of an experimental realization have been calculated. Further improvements using laser-cooled and trapped atoms are discussed.
Diffraction of atoms by lasers is a very important tool for matter wave optics. Although the process is well understood, the phase shifts induced by this diffraction process are not well known. In this paper, we make analytical calculations of these phase shifts in some simple cases and use these results to model the contrast interferometer recently built by Pritchard and co-workers. We thus show that the values of the diffraction phases are large and that they probably contribute to the phase noise observed in this experiment.