Classical and quantum modes of coupled Mathieu equations

Landa, H.; Drewsen, Michael; Reznik, Benni; Retzker, Alex

doi:10.1088/1751-8113/45/45/455305

Cited by 41 publications

(77 citation statements)

References 35 publications

(69 reference statements)

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“…Equation (32) is a linear differential equation with periodic coefficients and therefore amenable to treatment using Floquet theory, which we briefly review here. For more details, see [72] and references therein. For the Newtonian problem with f degrees of freedom ( f = 3N for N ions in three dimensions), the corresponding Floquet problem is stated in terms of coordinates in 2 f -dimensional phase space by the definitions…”

Section: The Floquet Problemmentioning

confidence: 99%

“…Excluding perhaps isolated values of β (and atypically in the a-q parameter space), all matrices that are inverted in the above expressions will be invertible, and while employing the algorithm in practice, the invertibility of the matrices is, of course, easily verified at each step. In section 5 we use, in fact, a generalization of the above expansion [72] which includes also the next Fourier harmonic (cos 4t, omitted from equation (30)).…”

Section: Solution Using An Expansion In Infinite Continued Matrix Invmentioning

confidence: 99%

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Modes of oscillation in radiofrequency Paul traps

et al. 2012

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We examine the time-dependent dynamics of ion crystals in radiofrequency traps. The problem of stable trapping of general threedimensional crystals is considered and the validity of the pseudopotential approximation is discussed. We analytically derive the micromotion amplitude of the ions, rigorously proving well-known experimental observations. We use a recently proposed method to find the modes that diagonalize the linearized time-dependent dynamical problem. This allows one to obtain explicitly the ('Floquet-Lyapunov') transformation to coordinates of decoupled linear oscillators. We demonstrate the utility of the method by analyzing the modes of a small 'peculiar' crystal in a linear Paul trap. The calculations can be readily generalized to multispecies ion crystals in general multipole traps, and time-dependent quantum wavefunctions of ion oscillations in such traps can be obtained.numerical study of a small crystal in a linear Paul trap, and conclude the paper in section 6 with comments on directions for further applications and research. Overview of Paul trappingThe first crystallization of a cloud of charged aluminum microparticles has been reported in [42]. Wigner crystals of 2 to approximately 100 trapped ions in a Paul trap were reported in [9,10] and further investigated in [43] both experimentally and numerically. The simulations included rf trapping, Coulomb interaction, laser cooling and random noise. Depending on trap parameters, the ions were found to equilibrate either as an apparently chaotic cloud or in an ordered structure. The latter is defined as the 'crystal' solution when it is a simple limit cycle, with the ions oscillating at the rf frequency about well-defined average points. The transition between the two phases has been investigated, it was shown that both phases can coexist and hysteresis in the transition has been observed [43].The motion of two ions in the Paul trap has been investigated in detail in various publications [44][45][46][47][48][49][50][51][52][53]. In addition to the aforementioned phases, frequency-locked periodic attractors (where the nonlinearity pulls the motional frequencies into integral fractions of the external rf frequency) were found in numerical simulations and experiments. These solutions are different from the crystal in that the ions move in extended (closed) orbits in the trap, whose period is a given multiple of the rf period. However, many of these frequency-locked solutions are unstable, especially those of a large period, and perturbations such as those coming from the nonlinearity of the laser-cooling mechanism tend to destroy them.Despite the large amplitude motion, the frequency-locked solutions, being periodic, are of course not chaotic. However, even in the presence of cooling, some solutions in the ion trap may behave chaotically for exponentially long times. The authors of [54] suggest that, eventually, all trajectories settle at frequency-locked attractors, at least for two ions at a = 0. Numerical simulations and experiments with more ion...

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Section: The Floquet Problemmentioning

confidence: 99%

Section: Solution Using An Expansion In Infinite Continued Matrix Invmentioning

confidence: 99%

Modes of oscillation in radiofrequency Paul traps

et al. 2012

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“…Moreover, our formalism can be directly applied to larger chains; preliminary studies show for certain parameter choices unexpected features in the scaling with the number of ions, motivating further examinations. In addition, our approach can be extended by taking the effects of the micromotion on the normal modes into account [33,34]. The analysis in this work provides the basis for investigations on the onset of thermalization in closed quantum systems [35,36], as well as studies of non-Markovian behavior in ion Coulomb crystals [37,38].…”

Section: Discussionmentioning

confidence: 99%

Quantum quenches of ion Coulomb crystals across structural instabilities. II. Thermal effects

2013

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We theoretically analyze the efficiency of a protocol for creating mesoscopic superpositions of ion chains, described in Baltrusch et al. [Phys. Rev. A 84, 063821 (2011)], as a function of the temperature of the crystal. The protocol makes use of state-dependent forces, so that a coherent superposition of the electronic states of one ion evolves into an entangled state between the chain's internal and external degrees of freedom. Ion Coulomb crystals are well isolated from the external environment and should therefore experience a coherent, unitary evolution, which follows the quench and generates structural Schrödinger-cat-like states. The temperature of the chain, however, introduces a statistical uncertainty in the final state. We characterize the quantum state of the crystal by means of the visibility of Ramsey interferometry performed on one ion of the chain and determine its decay as a function of the crystal's initial temperature. This analysis allows one to determine the conditions on the chain's initial state in order to efficiently perform the protocol.

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“…Notice that this set of coefficients had been found before by us [34,35] and others. [40,41] Nevertheless, these coefficients themselves are insufficient because they only permit to write the evolution operator in the form U A (see Equation (4)) whereas the calculation of H e requires the evolution operator to be written in the form of U B (see Equation (5)). Yet we need U B to find H e .…”

Section: Example 1: Harmonic Oscillator With Time-dependent Frequencymentioning

confidence: 99%

Method for Finding the Exact Effective Hamiltonian of Time‐Driven Quantum Systems

et al. 2019

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Time-driven quantum systems are important in many different fields of physics as cold atoms, solid state, optics, etc. Many of their properties are encoded in the time evolution operator or the effective Hamiltonian. Finding these operators usually requires very complex calculations that often involve some approximations. To perform this task, a systematic scheme that can be cast in the form of a symbolic computational algorithm is presented. It is suitable for periodic and non-periodic potentials and, for convoluted systems, can also be adapted to yield numerical solutions. The method exploits the structure of the associated Lie group and a decomposition of the evolution operator on each group generator. To illustrate the use of the method, five examples are provided: harmonic oscillator with time-dependent frequency (Paul trap), modulated optical lattice, time-driven quantum oscillator, a step-wise driving of a free particle, and the non-periodic Caldirola-Kanai Hamiltonian. To the extent of the authors' knowledge, whereas the exact form of Paul trap's evolution operator is well known, its effective Hamiltonian was until now unknown. The remaining four examples accurately reproduce previous results.

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Classical and quantum modes of coupled Mathieu equations

Cited by 41 publications

References 35 publications

Modes of oscillation in radiofrequency Paul traps

Modes of oscillation in radiofrequency Paul traps

Quantum quenches of ion Coulomb crystals across structural instabilities. II. Thermal effects

Method for Finding the Exact Effective Hamiltonian of Time‐Driven Quantum Systems

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