Planck scale corrections to the harmonic oscillator, coherent, and squeezed states

Bosso, Pasquale; Das, Saurya; Mann, Robert B.

doi:10.1103/physrevd.96.066008

Cited by 41 publications

(59 citation statements)

References 35 publications

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“…One usually assumes γ " 1{pM Pl cq, where M Pl is the Planck mass. Furthermore, the same parametrization was proved useful in perturbation theory [16]. Finally, notice that (1) represents an effective model giving rise to a minimal length of order " γ.…”

Section: Introductionmentioning

confidence: 87%

“…The two formalism are in fact usually assumed to be equivalent. More specifically, when GUP is considered in the quantum domain, the usual tools of quantum mechanics are implemented, rooted in the Hamiltonian formalism (see, for example also [15][16][17]). On the other hand, for a description of classical systems or in quantum field theory, the machinery of the Lagrangian formalism is preferred (see, for example [18][19][20]).…”

Section: Introductionmentioning

confidence: 99%

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Rigorous Hamiltonian and Lagrangian analysis of classical and quantum theories with minimal length

Bosso

2018

Phys. Rev. D

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GUP is a phenomenological model aimed for a description of a minimal length in quantum and classical systems. However, the analysis of problems in classical physics is usually approached preferring a different formalism than the one used for quantum systems, and vice versa. Potentially, the two approaches can result in inconsistencies. Here, we eliminate such inconsistencies proposing particular meanings and relations between the variables used to describe physical systems, resulting in a precise form of the Legendre transformation. Furthermore, we introduce two different sets of canonical variables and the relative map between them. These two sets allow for a complete and unambiguous description of classical and quantum systems. * pasquale.bosso@uleth.ca arXiv:1804.08202v2 [gr-qc]

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Rigorous Hamiltonian and Lagrangian analysis of classical and quantum theories with minimal length

Bosso

2018

Phys. Rev. D

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“…We start our consideration with the simple premise that the modification of the fundamental commutator for a harmonic oscillator is equivalent to the nonlinear modification of the Hamiltonian by means of the perturbative transformation of momentum,p →p − β 0p 3 /(3M 2 p c 2 ), which restores the canonic commutator, [x,p] = i , at the expense of adding the non-linear term to the Hamiltonian of the resonator:Ĥ →Ĥ 0 + ∆Ĥ = p 2 /2m + mΩ 2 0x 2 /2 + β 0p 4 / 3m(M p c) 2 . Such non-linear correction results in the dependence of the oscillator resonance frequency on its energy [7,9,18]. The dynamics of the system can be described by a well known Duffing oscillator model characterized by amplitude dependence of the resonance frequency, i.e.…”

Section: Theorymentioning

confidence: 99%

Testing the generalized uncertainty principle with macroscopic mechanical oscillators and pendulums

et al. 2019

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Recent progress in observing and manipulating mechanical oscillators at quantum regime provides new opportunities of studying fundamental physics, for example to search for low energy signatures of quantum gravity. For example, it was recently proposed that such devices can be used to test quantum gravity effects, by detecting the change in the [x,p] commutation relation that could result from quantum gravity corrections. We show that such a correction results in a dependence of a resonant frequency of a mechanical oscillator on its amplitude, which is known as amplitudefrequency effect. By implementing of this new method we measure amplitude-frequency effect for 0.3 kg ultra high-Q sapphire split-bar mechanical resonator and for ∼ 10 −5 kg quartz bulk acoustic wave resonator. Our experiments with sapphire resonator have established the upper limit on quantum gravity correction constant of β0 to not exceed 5.2 × 10 6 , which is factor of 6 better than previously measured. The reasonable estimates of β0 from experiments with quartz resonators yields β0 < 4×10 4 . The data sets of 1936 measurement of physical pendulum period by Atkinson [1] could potentially lead to significantly stronger limitations on β01. Yet, due to the lack of proper pendulum frequency stability measurement in these experiments the exact upper bound on β0 can not be reliably established. Moreover, pendulum based systems only allow to test a specific form of the modified commutator that depends on the mean value of momentum. The electro-mechanical oscillators to the contrary enable testing of any form of generalized uncertainty principle directly due to a much higher stability and higher degree of control.

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“…It was in fact shown that the Generalized Uncertainty Principle (GUP) gives rise to small Planckscale dependent terms in non-relativistic and relativistic Hamiltonians. These terms can nevertheless have potential experimental implications [1][2][3]. In particular, it was shown that there were p 3 and p 4 (p " momentum, q " position) Planck-scale corrections to the quantum Harmonic Oscillator (HO).…”

Section: Introductionmentioning

confidence: 99%

“…Such terms affect energy eigenvalues and eigenstates, that can be studied using standard perturbation techniques [4]. In a recent paper [3], it was shown however that an alternative approach can be pursued. Specifically, raising and lowering (or ladder) operators for the perturbed HO can be defined directly.…”

Section: Introductionmentioning

confidence: 99%

Generalized ladder operators for the perturbed harmonic oscillator

Bosso

Das

2018

Annals of Physics

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In this paper, we construct corrections to the raising and lowering (i.e. ladder) operators for a quantum harmonic oscillator subjected to a polynomial type perturbation of any degree and to any order in perturbation theory. We apply our formalism to a couple of examples, namely q and p 4 perturbations, and obtain the explicit form of those operators. We also compute the expectation values of position and momentum for the above perturbations. This construction is essential for defining coherent and squeezed states for the perturbed oscillator. Furthermore, this is the first time that corrections to ladder operators for a harmonic oscillator with a generic perturbation and to an arbitrary order of perturbation theory have been constructed.

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Planck scale corrections to the harmonic oscillator, coherent, and squeezed states

Cited by 41 publications

References 35 publications

Rigorous Hamiltonian and Lagrangian analysis of classical and quantum theories with minimal length

Rigorous Hamiltonian and Lagrangian analysis of classical and quantum theories with minimal length

Testing the generalized uncertainty principle with macroscopic mechanical oscillators and pendulums

Generalized ladder operators for the perturbed harmonic oscillator

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