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Suplemento de la Revista Mexicana de Física

Numerical implementation of a Mach-Zehnder interferometer for Bose-Einstein condensates

Jessica Gil-Londoño, Gustavo Marín-Alvarado, Karen Rodríguez

Published Aug 22, 2020• 0 citations
Publisher

Abstract

We numerically implement a Mach-Zehnder interferometer, where the coherence and oscillatory properties of Bose-Einstein condensates are explored and the system is modeled by the Gross-Pitaevskii equation. Several time-dependent external trapping potentials were engineered seeking the adiabatic regime which is quantified using fidelity measurements with respect to the actual ground-state of the trap. The dynamics of both conjugate variables, namely density and phase of the matter-wave function, are shown. Moreover, the density and fidelity profiles of the system are presented when the phase-shifter is switching-on and -off, being found in the presented profiles that the system exhibits three different regimes during the recombination stage among them even an orthogonal BEC to the original one is obtained. We achieve the numerical solution through an adequate implementation of the finite-difference method for the spatial discretization and a Runge-Kutta method for the time evolution.

DOI
10.31349/suplrevmexfis.1.3.31

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We numerically implement a Mach-Zehnder interferometer, where the coherence and oscillatory properties of Bose-Einstein condensates are explored and the system is modeled by the Gross-Pitaevskii equation. Several time-dependent external trapping potentials were engineered seeking the adiabatic regime which is quantified using fidelity measurements with respect to the actual ground-state of the trap. The dynamics of both conjugate variables, namely density and phase of the matter-wave function, are shown. Moreover, the density and fidelity profiles of the system are presented when the phase-shifter is switching-on and -off, being found in the presented profiles that the system exhibits three different regimes during the recombination stage among them even an orthogonal BEC to the original one is obtained. We achieve the numerical solution through an adequate implementation of the finite-difference method for the spatial discretization and a Runge-Kutta method for the time evolution.
More than two decades ago a new state of matter called Bose-Einstein Condensation was proved experimentally although predicted long before in 1924-25.At extremely low temperatures and controlled densities, this exotic state can be achieved where all atoms share the same energy and linear momentum similarly as the light behaves in a laser.Therefore, it is interesting to analyze the matterwave counterparts for the nowadays well-known optical devices.Here, we present a numerical implementation of a Mach-Zehnder interferometer which we engineered by means of time-dependent external trapping potentials.We use an appropriate implementation of the finite-difference method for the spatial discretization and the Runge-Kutta technique for the time evolution.The dynamics of both conjugate variables, namely density and phase of the matter-wave functions are shown.Our aim is also to show that this kind of problems and techniques can be addressed in standard courses of Quantum Mechanics and Computational Physics.
Particle-wave duality enables the construction of interferometers for matter waves, which complement optical interferometers in precision measurement devices. This requires the development of atom-optics analogues to beam splitters, phase shifters and recombiners. Integrating these elements into a single device has been a long-standing goal. Here we demonstrate a full Mach–Zehnder sequence with trapped Bose–Einstein condensates confined on an atom chip. Particle interactions in our Bose–Einstein condensate matter waves lead to a nonlinearity, absent in photon optics. We exploit it to generate a non-classical state having reduced number fluctuations inside the interferometer. Making use of spatially separated wave packets, a controlled phase shift is applied and read out by a non-adiabatic matter-wave recombiner. We demonstrate coherence times a factor of three beyond what is expected for coherent states, highlighting the potential of entanglement as a resource for metrology. Our results pave the way for integrated quantum-enhanced matter-wave sensors. Atom interferometers exploit wave-particle duality and can be used as sensitive measurement devices. Berrada et al.present a Mach–Zehnder interferometer for Bose–Einstein condensates trapped on an atom chip and demonstrate enhanced performance using non-classical states.
We present a generalized model for the evolution of atomic wave-packets in matter-wave interferometers. This method provides an efficient tool for analyzing the performance of atomic interferometers using atom clouds prepared in a trap as a Bose-Einstein condensate (BEC) or as a thermal ensemble. Predictions of the model for dynamic properties such as wave-packet size and phase are in excellent agreement with explicit numerical solutions of the non-linear Gross-Pitaevskii equations and enable fast calculations of interferometric performance in regimes where full numerical solutions become impractical. As a starting point, the static Thomas-Fermi (TF) approximation for a BEC in a harmonic trap is generalized to the whole range of atom-atom interaction strengths: from non-interacting atoms (low densities) to the standard TF limit (high atomic densities, as long as the condensate approximation still holds). In particular, this generalization allows a good estimation of atomic cloud properties along the transition from a three-dimensional to a quasi-one-dimensional BEC in an elongated trap. We then develop a theoretical model of wave-packet evolution in time-dependent conditions. The model is applicable for a wide range of dynamical problems involving evolution in time-dependent potentials and in a changing mean-field atomic repulsion due to splitting and separation of wave-packets. We use the model for studying two effects that influence interferometric coherence: imperfect spatial recombination in a two-state interferometer (the so-called "Humpty-Dumpty effect") and phase diffusion due to number uncertainty in the two interferometer arms, which was previously studied thoroughly only for interferometric schemes where the BECs in the two arms stay trapped (for example, in a double-well potential).
We analyze how the contrast of a quasi-one-dimensional Bose-Einstein-condensation (BEC)-based atomic interferometer is affected by the number density of atoms. In our numerical contrast of a Mach-Zehnder-type BEC-based interferometer a regularly spaced oscillatory structure emerges as a function of both the number density and the recombination time. We attribute this effect to the phase singularities in Husimi representation of the overlapping BEC wave packets caused by the residual nonuniform velocity distribution.
We present the application of a fast, explicit time-marching scheme for the solution of the Gross-Pitaevskii equation in cylindrical geometry. The scheme is validated on simple analytical tests and demonstrated for two situations of physical interest in experiments on the Bose-Einstein condensation (BEC) of trapped alkali-metal vapors. It is tested by reproducing known results on the free expansion of a BEC after removing a cylindrical trap, and it is then used to address the formation of matter-wave pulses that result from gravity-induced transport of a condensate in an optical potential.