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New Journal of Physics

A fidelity treatment of near-resonant states in the atom-optics kicked rotor

Peter D. McDowall, Andrew J. Hilliard, M. McGovern, Tzahi Grünzweig, Mikkel F. Andersen

Published Dec 21, 2009• 24 citations
Publisher

Abstract

We investigate the dynamics of the atom-optics δ-kicked rotor in the vicinity of quantum resonance. Although small deviations from resonant conditions lead to a negligible change in the momentum space probability density, they lead to a significant relative phase change between the different momentum states taking part in the dynamics. By adding a tailored pulse to the kicked rotor pulse sequence, one can measure the overlap between the resonant state and any other state, i.e. perform a fidelity measurement. Using this sequence, we predict a narrow peak around quantum resonance with a width that scales as 1/ N 3 with N being the number of pulses in the kicked rotor sequence. This method may be of interest to precision measurements, such as h / M .

DOI
10.1088/1367-2630/11/12/123021

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•Jan 1, 2006•20 citations
We investigate the dynamics of the atom-optics δ-kicked rotor in the vicinity of quantum resonance. Although small deviations from resonant conditions lead to a negligible change in the momentum space probability density, they lead to a significant relative phase change between the different momentum states taking part in the dynamics. By adding a tailored pulse to the kicked rotor pulse sequence, one can measure the overlap between the resonant state and any other state, i.e. perform a fidelity measurement. Using this sequence, we predict a narrow peak around quantum resonance with a width that scales as 1/ N 3 with N being the number of pulses in the kicked rotor sequence. This method may be of interest to precision measurements, such as h / M .
The study of quantum resonances in the chaotic atom-optics kicked rotor system is of interest from two different perspectives. In quantum chaos, it marks out the regime of resonant quantum dynamics in which the atomic cloud displays ballistic mean energy growth due to coherent momentum transfer. Secondly, the sharp quantum resonance peaks are useful in the context of measurement of Talbot time, one of the parameter that helps in precise measurement of fine structure constant. Most of the earlier works rely on fidelity based approach and have proposed Talbot time measurement through experimental determination of the momentum space probability density of the periodically kicked atomic cloud. Fidelity approach has the disadvantage that phase reversed kicks need to be imparted as well which potentially leads to dephasing. In contrast to this, in this work, it is theoretically shown that, without manipulating the kick sequences, the quantum resonances through position space density can be measured more accurately and is experimentally feasible as well.
Abstract This experimental work demonstrates the importance of finite-width effects in the evolution of a quantum system, where the results deviate considerably from the plane wave approximation even for an initial state with a very narrow momentum width i.e a Bose–Einstein condensate. The system under consideration is an atom optics δ -kicked rotor for which a fidelity based measurement has been proposed to possess a rapid scaling of sensitivity ( N −3 ) with pulse number N . Although attractive, we demonstrate that this scaling does not hold in the regime where the momentum selectivity of the pulse sequence becomes significantly smaller than the momentum width of the initial state. Additionally, the momentum distribution post kicking shows a lattice-phase dependent intra-order and inter-order asymmetry in the diffracted orders. The intra-order asymmetry, in which no net momentum current is present, is a previously unreported type of effect. For a two pulse case, the inter-order asymmetry signal is found to be about five times more sensitive to the resonance than the initial state fidelity. Both of these asymmetries provide a zero-crossing signal which can be used to diagnose any undesirable lattice phase offset at resonant and off-resonant pulse periods.
We experimentally investigate the sub-Fourier behavior of a δ-kicked-rotor resonance by performing a measurement of the fidelity or overlap of a Bose-Einstein condensate exposed to a periodically pulsed standing wave. The temporal width of the fidelity resonance peak centered at the Talbot time and zero initial momentum exhibits an inverse cube pulse number (1/N3)-dependent scaling compared to a 1/N2 dependence for the mean energy width at the same resonance. A theoretical analysis shows that for an accelerating potential the width of the resonance in acceleration space depends on 1/N3, a property which we also verify experimentally. Such a sub-Fourier effect could be useful for high precision gravity measurements.
The effect of pulse train noise on the quantum resonance peaks of the atom optics kicked rotor is investigated experimentally. Quantum resonance peaks in the late time mean energy of the atoms are found to be surprisingly robust against all levels of noise applied to the kicking amplitude, while even small levels of noise on the kicking period lead to their destruction. The robustness to amplitude noise of the resonance peak and of the fall-off in mean energy to either side of this peak are explained in terms of the occurrence of stable, epsilon classical dynamics [Nonlinearity 16, 1381 (2003)]] around each quantum resonance.
We show that the quantum fidelity is accessible to cold atom experiments for a large class of evolutions in periodical potentials, properly taking into account the experimental initial conditions of the atomic ensemble. We prove analytically that, at the fundamental quantum resonances of the atom-optics kicked rotor, the fidelity saturates at a constant, time-independent value after a small number of kicks. The latter saturation arises from the bulk of the atomic ensemble, whilst for the resonantly accelerated atoms the fidelity is predicted to decay slowly according to a power law.
We show that the quantum fidelity is accessible to cold atom experiments for a large class of evolutions in periodical potentials, properly taking into account the experimental initial conditions of the atomic ensemble. We prove analytically that, at the fundamental quantum resonances of the atom-optics kicked rotor, the fidelity saturates at a constant, time-independent value after a small number of kicks. The latter saturation arises from the bulk of the atomic ensemble, whilst for the resonantly accelerated atoms the fidelity is predicted to decay slowly according to a power law.