Propagation of arbitrary initial wave packets in a quantum parametric oscillator: Instability zones for higher order moments

Biswas, Subhadip; Chattopadhyay, Rohitashwa; Bhattacharjee, Jayanta K.

doi:10.1016/j.physleta.2018.03.011

Cited by 5 publications

(5 citation statements)

References 11 publications

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“…The dynamics of the variance as shown in Eq. (2.13) has been investigated in [4]- [5] for µ = 0 (periodic modulation). We note that from Eqs.…”

Section: The Equation Of Motionmentioning

confidence: 99%

“…It is consequently important to know the zones where these divergences occur. Recent work [4]- [6] has established quite clearly that the variance and the mean diverge in the same regions of the frequency-amplitude plane leading to the situation that if the particle is trapped its energy will be finite..…”

mentioning

confidence: 99%

“…While references [4] and [5] have dealt with the variance, it should be noted that reference [6] has dealt with a more physical quantity which is the energy expectation value and shown that the region of divergence of mean position and the average energy are the same. We show the connection between the variance and the average energy in Sec II.…”

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confidence: 99%

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Instability Zones in the Dynamics of a Quantum Mechanical Quasiperiodic Parametric Oscillator

Biswas,

Chowdhury,

Bhattacharjee

2019

Preprint

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Quasi-periodically driven quantum parametric oscillators have been the subject of several recent investigations. Here we show that for such oscillators, the instability zones of the mean position and variance (alternatively the mean energy) for a time developing wave packet are identical for the strongest resonance in the three-dimensional parameter space of the quasi-periodic modulation as it is for the two-dimensional parameter space of the periodic modulations.

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“…The dynamics of the variance as shown in Eq. (2.13) has been investigated in [4]- [5] for µ = 0 (periodic modulation). We note that from Eqs.…”

Section: The Equation Of Motionmentioning

confidence: 99%

mentioning

confidence: 99%

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Instability Zones in the Dynamics of a Quantum Mechanical Quasiperiodic Parametric Oscillator

Biswas,

Chowdhury,

Bhattacharjee

2019

Preprint

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“…In the quantum case, the corresponding dynamics should in principle be the dynamics of an initial wave packet and there have been several attempts of late [12][13][14] to study quantum dynamics in the presence of barriers and quantum resonances. Here we adopt a different approach of looking at the dynamics following a generalization of Erhenfest's theorem to account for the quantum fluctuations [15]. Consequently we find that what would be a bound orbit inside the central dip of the potential becomes an escaping orbit due to quantum fluctuations.…”

Section: Introductionmentioning

confidence: 99%

From periodically driven double wells to volcano potentials: Quantum dynamics

Paul,

Chawla,

Das

et al. 2018

Preprint

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We consider the dynamics of a particle confined in a double well potential which is subjected to a periodic drive. In the case of deep and well separated wells, we find that by adjusting the parameters of the drive we can generate, to a very good approximation, a volcano potential. The quantum dynamics in this volcano potential is studied by a variation of what can be called a generalized Ehrenfest's theorem. We find that the coupling of the mean position and the width of the wave packet in this dynamics causes the particle to escape from the central well in accordance with the fact that the volcano potential only supports resonance states.

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“…In this situation, Ehrenfest's theorem is very useful since it provides exact equations for all the moments. Each moment gives rise to a nonautonomous dynamical [28] system, and this very physical problem gives rise to the whole set of interesting mathematical systems which are worth studying in their own right. In this work, we will focus on the mathematical aspects of the dynamical systems arising from Ehrenfest's theorem for the moments.…”

Section: Introductionmentioning

confidence: 99%

On the properties of a class of higher-order Mathieu equations originating from a parametric quantum oscillator

Biswas

Bhattacharjee

2019

Nonlinear Dyn

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Evolution in time of an arbitrary initial state for a parametrically driven quantum oscillator is an interesting problem since there exist regions in parameter space (defined by the amplitude and frequency of the driving) where the moments of the probability distribution can diverge in time. While the first moment satisfies a Mathieu equation, the higher-order moments follow Mathieu like equations of order greater than two. It is not very often that a physical problem gives rise to higher-order Mathieu equations. Hence, we give a detailed study of the different stability zones associated with the parametric quantum oscillator, using perturbative techniques traditionally associated with the Mathieu equation. We verify our results by numerical analysis, thus demonstrating that for the higher-order Mathieu equations, the traditional perturbation theory methods give a consistent account of the stability zones.

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Propagation of arbitrary initial wave packets in a quantum parametric oscillator: Instability zones for higher order moments

Cited by 5 publications

References 11 publications

Instability Zones in the Dynamics of a Quantum Mechanical Quasiperiodic Parametric Oscillator

Instability Zones in the Dynamics of a Quantum Mechanical Quasiperiodic Parametric Oscillator

From periodically driven double wells to volcano potentials: Quantum dynamics

On the properties of a class of higher-order Mathieu equations originating from a parametric quantum oscillator

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