Coherent states for the time dependent harmonic oscillator: the step function

The correspondence between classical and quantum invariants is established. The Ermakov Lewis quantum invariant of the time dependent harmonic oscillator is translated from the coordinate and momentum operators into amplitude and phase operators. In doing so, Turski's phase operator as well as Susskind-Glogower operators are generalized to the time dependent harmonic oscillator case. A quantum derivation of the Manley-Rowe relations is shown as an example.

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“…This assertion is consistent with the transformation that relates the invariant and the time dependent Hamiltonian [3] …”

Section: Creation and Annihilation Operatorssupporting

confidence: 87%

Amplitude and phase representation of quantum invariants for the time-dependent harmonic oscillator

Fernández‐Guasti

Moya‐Cessa

2003

Phys. Rev. A

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“…The final result of a modification in the initial frequency produces squeezed coherent states [11], as it is described by Dutra [12].…”

Section: Comparison Of the Standard And The Matrix Methodsmentioning

confidence: 96%

Alternative analysis to perturbation theory in quantum mechanics

2012

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We develop an alternative approach to time independent perturbation theory in non-relativistic quantum mechanics. The method developed has the advantage to provide in one operation the correction to the energy and to the wave function, additionally we can analyze the time evolution of the system. To verify our results, we apply our method to the harmonic oscillator perturbed by a quadratic potential. An alternative form of the Dyson series, in matrix form instead of integral form, is also obtained.

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“…It has been shown already that for low intensities it is possible also to consider the ion micromotion [26], and by using Ermakov-Lewis invariant methods [30,31,32] it was possible to linearize the ionlaser Hamiltonian when the micromotion was included [44]. Here, we follow Zúñiga et al [45] and show how it is possible to solve the ion-laser interaction in different regimes, including high intensity and medium intensity.…”

Section: Solution In Different Regimes: Dispersive Hamiltoniansmentioning

confidence: 96%

Ion–laser interactions: The most complete solution

Moya‐Cessa¹,

Soto‐Eguibar²,

Vargas-Martínez³

et al. 2012

Physics Reports

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Trapped ions are considered one of the best candidates to perform quantum information processing. By interacting them with laser beams they are, somehow, easy to manipulate, which makes them an excellent choice for the production of nonclassical states of their vibrational motion, the reconstruction of quasiprobability distribution functions, the production of quantum gates, etc. However, most of these effects have been produced in the so-called low intensity regime, this is, when the Rabi frequency (laser intensity) is much smaller than the trap frequency. Because of the possibility to produce faster quantum gates in other regimes it is of importance to study this system in a more complete manner, which is the motivation for this contribution. We start by studying the way ions are trapped in Paul traps in and review the basic mechanisms of trapping. Then we show how the problem may be completely solved for trapping states; i.e., we find (exact) eigenstates of the full Hamiltonian. We show how in the low intensity regime Jaynes-Cummings and anti-Jaynes-Cummings interactions may be obtained, without using the rotating wave approximation and analyze the medium and high intensity regimes were dispersive Hamiltonians are produced. The traditional approach (low intensity regime) is also studied and used for the generation of non-classical states of the vibrational of the vibrational wavefunction. In particular, we show how to add and subtract vibrational quanta to an initial state, how to produce specific superpositions of number states and how to generate NOON states for the two-dimensional vibration of the ion. It is also shown how squeezing may be measured. The time dependent problem is studied by using Lewis-Ermakov methods, we give a solution to the problem when the time dependence of the trap is considered and also analyze an specific (artificial) time dependence that produces squeezing of the initial vibrational wave function. A way to mimic the ion-laser interaction via classical optics is also introduced.

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Coherent states for the time dependent harmonic oscillator: the step function

Cited by 48 publications

References 12 publications

Amplitude and phase representation of quantum invariants for the time-dependent harmonic oscillator

Amplitude and phase representation of quantum invariants for the time-dependent harmonic oscillator

Alternative analysis to perturbation theory in quantum mechanics

Ion–laser interactions: The most complete solution

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