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The quantum kicked rotor (QKR) map is embedded into a continuous unitary transformation generated by a time-independent quasi-Hamiltonian. In some vicinity of a quantum resonance of order $q$, we relate the problem to the {it regular} motion along a circle in a $(q^2-1)$-component inhomogeneous magnetic field of a quantum particle with $q$ intrinsic degrees of freedom described by the $SU(q)$ group. This motion is in parallel with the classical phase oscillations near a non-linear resonance.
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The quantum kicked rotor (QKR) driven by $d$ incommensurate frequencies realizes the universality class of $d$-dimensional disordered metals. For $d>3$, the system exhibits an Anderson metal-insulator transition which has been observed within the framework of an atom optics realization. However, the absence of genuine randomness in the QKR reflects in critical phenomena beyond those of the Anderson universality class. Specifically, the system shows strong sensitivity to the algebraic properties of its effective Planck constant $tilde h equiv 4pi /q$. For integer $q$, the system may be in a globally integrable state, in a `super-metallic configuration characterized by diverging response coefficients, Anderson localized, metallic, or exhibit transitions between these phases. We present numerical data for different $q$-values and effective dimensionalities, with the focus being on parameter configurations which may be accessible to experimental investigations.
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.
We study the dynamics of the many-body atomic kicked rotor with interactions at the mean-field level, governed by the Gross-Pitaevskii equation. We show that dynamical localization is destroyed by the interaction, and replaced by a subdiffusive behavior. In contrast to results previously obtained from a simplified version of the Gross-Pitaevskii equation, the subdiffusive exponent does not appear to be universal. By studying the phase of the mean-field wave function, we propose a new approximation that describes correctly the dynamics at experimentally relevant times close to the start of subdiffusion, while preserving the reduced computational cost of the former approximation.
We propose two experimentally feasible methods based on atom interferometry to measure the quantum state of the kicked rotor.
We develop the Wigner phase space representation of a kicked particle for an arbitrary but periodic kicking potential. We use this formalism to illustrate quantum resonances and anti--resonances.
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